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|---|---|---|---|---|---|---|
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
this :
coequalizer.π (SheafedSpace.forgetToPresheafedSpace.map f.val) (SheafedSpace.forgetToPresheafedSpace.map g.val) ≫
(PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom =
SheafedSpace.forgetToPresheafedSpace.map (coequalizer.π f.val g.val)
⊢ IsLocalRingHom
(((PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c.app (op U) ≫
(PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit
(parallelPair (SheafedSpace.forgetToPresheafedSpace.map f.val)
(SheafedSpace.forgetToPresheafedSpace.map g.val))
((Opens.map (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.base).obj
(op U).unop)).hom ≫
limit.π
(PresheafedSpace.componentwiseDiagram
(parallelPair (SheafedSpace.forgetToPresheafedSpace.map f.val)
(SheafedSpace.forgetToPresheafedSpace.map g.val))
((Opens.map (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.base).obj
(op U).unop))
(op WalkingParallelPair.one)) ≫
Y.presheaf.map
(eqToHom
(_ :
(Opens.map
(coequalizer.π (SheafedSpace.forgetToPresheafedSpace.map f.val)
(SheafedSpace.forgetToPresheafedSpace.map g.val) ≫
(PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom).base).op.obj
(op U) =
(Opens.map (SheafedSpace.forgetToPresheafedSpace.map (coequalizer.π f.val g.val)).base).op.obj (op U))))
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
|
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
|
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.145_0.tE6q65npbp8AX2g
|
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U))
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
this✝ :
coequalizer.π (SheafedSpace.forgetToPresheafedSpace.map f.val) (SheafedSpace.forgetToPresheafedSpace.map g.val) ≫
(PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom =
SheafedSpace.forgetToPresheafedSpace.map (coequalizer.π f.val g.val)
this : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c
⊢ IsLocalRingHom
(((PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c.app (op U) ≫
(PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit
(parallelPair (SheafedSpace.forgetToPresheafedSpace.map f.val)
(SheafedSpace.forgetToPresheafedSpace.map g.val))
((Opens.map (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.base).obj
(op U).unop)).hom ≫
limit.π
(PresheafedSpace.componentwiseDiagram
(parallelPair (SheafedSpace.forgetToPresheafedSpace.map f.val)
(SheafedSpace.forgetToPresheafedSpace.map g.val))
((Opens.map (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.base).obj
(op U).unop))
(op WalkingParallelPair.one)) ≫
Y.presheaf.map
(eqToHom
(_ :
(Opens.map
(coequalizer.π (SheafedSpace.forgetToPresheafedSpace.map f.val)
(SheafedSpace.forgetToPresheafedSpace.map g.val) ≫
(PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom).base).op.obj
(op U) =
(Opens.map (SheafedSpace.forgetToPresheafedSpace.map (coequalizer.π f.val g.val)).base).op.obj (op U))))
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
|
infer_instance
|
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.145_0.tE6q65npbp8AX2g
|
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U))
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
⊢ ⇑(coequalizer.π f.val g.val).base ⁻¹' (⇑(coequalizer.π f.val g.val).base '' (imageBasicOpen f g U s).carrier) =
(imageBasicOpen f g U s).carrier
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
|
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
|
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.187_0.tE6q65npbp8AX2g
|
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case e
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
⊢ ⇑f.val.base ≫ ⇑(coequalizer.π f.val g.val).base = ⇑g.val.base ≫ ⇑(coequalizer.π f.val g.val).base
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
·
|
ext
|
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
·
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.187_0.tE6q65npbp8AX2g
|
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case e.h
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
a✝ : (forget TopCat).obj ↑X.toPresheafedSpace
⊢ (⇑f.val.base ≫ ⇑(coequalizer.π f.val g.val).base) a✝ = (⇑g.val.base ≫ ⇑(coequalizer.π f.val g.val).base) a✝
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
|
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
|
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.187_0.tE6q65npbp8AX2g
|
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case e.h
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
a✝ : (forget TopCat).obj ↑X.toPresheafedSpace
⊢ (f.val ≫ coequalizer.π f.val g.val).base a✝ = (g.val ≫ coequalizer.π f.val g.val).base a✝
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
|
congr 2
|
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.187_0.tE6q65npbp8AX2g
|
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case e.h.e_a.e_self
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
a✝ : (forget TopCat).obj ↑X.toPresheafedSpace
⊢ f.val ≫ coequalizer.π f.val g.val = g.val ≫ coequalizer.π f.val g.val
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
|
exact coequalizer.condition f.1 g.1
|
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.187_0.tE6q65npbp8AX2g
|
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case h
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
⊢ IsColimit
(Cofork.ofπ ⇑(coequalizer.π f.val g.val).base
(_ : ⇑f.val.base ≫ ⇑(coequalizer.π f.val g.val).base = ⇑g.val.base ≫ ⇑(coequalizer.π f.val g.val).base))
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
·
|
apply isColimitCoforkMapOfIsColimit (forget TopCat)
|
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
·
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.187_0.tE6q65npbp8AX2g
|
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case h.l
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
⊢ IsColimit (Cofork.ofπ (coequalizer.π f.val g.val).base ?h.w)
case h.w
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
⊢ f.val.base ≫ (coequalizer.π f.val g.val).base = g.val.base ≫ (coequalizer.π f.val g.val).base
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
|
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
|
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.187_0.tE6q65npbp8AX2g
|
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case h.l.l
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
⊢ IsColimit (Cofork.ofπ (coequalizer.π f.val g.val) ?h.l.w)
case h.l.w
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
⊢ f.val ≫ coequalizer.π f.val g.val = g.val ≫ coequalizer.π f.val g.val
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
|
exact coequalizerIsCoequalizer f.1 g.1
|
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.187_0.tE6q65npbp8AX2g
|
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case H
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
⊢ ⇑f.val.base ⁻¹' (imageBasicOpen f g U s).carrier = ⇑g.val.base ⁻¹' (imageBasicOpen f g U s).carrier
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
·
|
suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
|
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
·
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.187_0.tE6q65npbp8AX2g
|
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
this : (Opens.map f.val.base).obj (imageBasicOpen f g U s) = (Opens.map g.val.base).obj (imageBasicOpen f g U s)
⊢ ⇑f.val.base ⁻¹' (imageBasicOpen f g U s).carrier = ⇑g.val.base ⁻¹' (imageBasicOpen f g U s).carrier
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by
|
injection this
|
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.187_0.tE6q65npbp8AX2g
|
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case H
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
⊢ (Opens.map f.val.base).obj (imageBasicOpen f g U s) = (Opens.map g.val.base).obj (imageBasicOpen f g U s)
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
|
delta imageBasicOpen
|
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.187_0.tE6q65npbp8AX2g
|
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case H
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
⊢ (Opens.map f.val.base).obj
(RingedSpace.basicOpen (toRingedSpace Y)
(let_fun this := ((coequalizer.π f.val g.val).c.app (op U)) s;
this)) =
(Opens.map g.val.base).obj
(RingedSpace.basicOpen (toRingedSpace Y)
(let_fun this := ((coequalizer.π f.val g.val).c.app (op U)) s;
this))
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
|
rw [preimage_basicOpen f, preimage_basicOpen g]
|
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.187_0.tE6q65npbp8AX2g
|
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case H
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
⊢ RingedSpace.basicOpen (toRingedSpace X)
((f.val.c.app
{
unop :=
{
unop :=
{ carrier := ⇑(coequalizer.π f.val g.val).base ⁻¹' ↑(op U).unop,
is_open' := (_ : IsOpen (⇑(coequalizer.π f.val g.val).base ⁻¹' ↑(op U).unop)) } }.unop })
(let_fun this := ((coequalizer.π f.val g.val).c.app (op U)) s;
this)) =
RingedSpace.basicOpen (toRingedSpace X)
((g.val.c.app
{
unop :=
{
unop :=
{ carrier := ⇑(coequalizer.π f.val g.val).base ⁻¹' ↑(op U).unop,
is_open' := (_ : IsOpen (⇑(coequalizer.π f.val g.val).base ⁻¹' ↑(op U).unop)) } }.unop })
(let_fun this := ((coequalizer.π f.val g.val).c.app (op U)) s;
this))
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
|
dsimp only [Functor.op, unop_op]
|
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.187_0.tE6q65npbp8AX2g
|
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case H
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
⊢ RingedSpace.basicOpen (toRingedSpace X)
((f.val.c.app
{
unop :=
{ carrier := ⇑(coequalizer.π f.val g.val).base ⁻¹' ↑U,
is_open' := (_ : IsOpen (⇑(coequalizer.π f.val g.val).base ⁻¹' ↑(op U).unop)) } })
(((coequalizer.π f.val g.val).c.app (op U)) s)) =
RingedSpace.basicOpen (toRingedSpace X)
((g.val.c.app
{
unop :=
{ carrier := ⇑(coequalizer.π f.val g.val).base ⁻¹' ↑U,
is_open' := (_ : IsOpen (⇑(coequalizer.π f.val g.val).base ⁻¹' ↑(op U).unop)) } })
(((coequalizer.π f.val g.val).c.app (op U)) s))
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
|
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
|
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.187_0.tE6q65npbp8AX2g
|
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case H
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
⊢ ((Opens.map (f.val ≫ coequalizer.π f.val g.val).base).op.obj (op U)).unop ⊓
RingedSpace.basicOpen (toRingedSpace X) (((g.val ≫ coequalizer.π f.val g.val).c.app (op U)) s) =
RingedSpace.basicOpen (toRingedSpace X) (((g.val ≫ coequalizer.π f.val g.val).c.app (op U)) s)
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
|
apply inf_eq_right.mpr
|
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.187_0.tE6q65npbp8AX2g
|
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case H
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
⊢ RingedSpace.basicOpen (toRingedSpace X) (((g.val ≫ coequalizer.π f.val g.val).c.app (op U)) s) ≤
((Opens.map (f.val ≫ coequalizer.π f.val g.val).base).op.obj (op U)).unop
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
|
refine' (RingedSpace.basicOpen_le _ _).trans _
|
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.187_0.tE6q65npbp8AX2g
|
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case H
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
⊢ ((Opens.map (g.val ≫ coequalizer.π f.val g.val).base).op.obj (op U)).unop ≤
((Opens.map (f.val ≫ coequalizer.π f.val g.val).base).op.obj (op U)).unop
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
|
rw [coequalizer.condition f.1 g.1]
|
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.187_0.tE6q65npbp8AX2g
|
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
⊢ IsOpen (⇑(coequalizer.π f.val g.val).base '' (imageBasicOpen f g U s).carrier)
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
|
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
|
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.216_0.tE6q65npbp8AX2g
|
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
⊢ IsOpen
(⇑(TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget CommRingCat) f.val g.val)) ∘
⇑(colimit.ι
(parallelPair ((SheafedSpace.forget CommRingCat).map f.val) ((SheafedSpace.forget CommRingCat).map g.val))
WalkingParallelPair.one) ⁻¹'
(⇑(coequalizer.π f.val g.val).base '' (imageBasicOpen f g U s).carrier))
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
|
erw [← coe_comp]
|
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.216_0.tE6q65npbp8AX2g
|
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
⊢ IsOpen
(⇑(colimit.ι
(parallelPair ((SheafedSpace.forget CommRingCat).map f.val) ((SheafedSpace.forget CommRingCat).map g.val))
WalkingParallelPair.one ≫
(PreservesCoequalizer.iso (SheafedSpace.forget CommRingCat) f.val g.val).hom) ⁻¹'
(⇑(coequalizer.π f.val g.val).base '' (imageBasicOpen f g U s).carrier))
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
|
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
|
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.216_0.tE6q65npbp8AX2g
|
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
⊢ IsOpen
(⇑((SheafedSpace.forget CommRingCat).map (coequalizer.π f.val g.val)) ⁻¹'
(⇑(coequalizer.π f.val g.val).base '' (imageBasicOpen f g U s).carrier))
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
|
dsimp only [SheafedSpace.forget]
|
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.216_0.tE6q65npbp8AX2g
|
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
⊢ IsOpen (⇑(coequalizer.π f.val g.val).base ⁻¹' (⇑(coequalizer.π f.val g.val).base '' (imageBasicOpen f g U s).carrier))
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
|
erw [imageBasicOpen_image_preimage]
|
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.216_0.tE6q65npbp8AX2g
|
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
⊢ IsOpen (imageBasicOpen f g U s).carrier
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
|
exact (imageBasicOpen f g U s).2
|
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.216_0.tE6q65npbp8AX2g
|
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
x : ↑(toTopCat Y)
⊢ IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val) x)
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
|
constructor
|
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.228_0.tE6q65npbp8AX2g
|
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case map_nonunit
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
x : ↑(toTopCat Y)
⊢ ∀ (a : ↑(PresheafedSpace.stalk (coequalizer f.val g.val).toPresheafedSpace ((coequalizer.π f.val g.val).base x))),
IsUnit ((PresheafedSpace.stalkMap (coequalizer.π f.val g.val) x) a) → IsUnit a
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
|
rintro a ha
|
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.228_0.tE6q65npbp8AX2g
|
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case map_nonunit
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
x : ↑(toTopCat Y)
a : ↑(PresheafedSpace.stalk (coequalizer f.val g.val).toPresheafedSpace ((coequalizer.π f.val g.val).base x))
ha : IsUnit ((PresheafedSpace.stalkMap (coequalizer.π f.val g.val) x) a)
⊢ IsUnit a
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
|
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
|
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.228_0.tE6q65npbp8AX2g
|
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case map_nonunit.intro.intro.intro
X Y : LocallyRingedSpace
f g : X ⟶ Y
U✝ : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s✝ : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U✝))
x : ↑(toTopCat Y)
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
hU : (coequalizer.π f.val g.val).base x ∈ U
s : (forget CommRingCat).obj ((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
ha :
IsUnit
((PresheafedSpace.stalkMap (coequalizer.π f.val g.val) x)
((TopCat.Presheaf.germ (coequalizer f.val g.val).toPresheafedSpace.presheaf
{ val := (coequalizer.π f.val g.val).base x, property := hU })
s))
⊢ IsUnit
((TopCat.Presheaf.germ (coequalizer f.val g.val).toPresheafedSpace.presheaf
{ val := (coequalizer.π f.val g.val).base x, property := hU })
s)
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
|
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
|
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.228_0.tE6q65npbp8AX2g
|
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case map_nonunit.intro.intro.intro
X Y : LocallyRingedSpace
f g : X ⟶ Y
U✝ : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s✝ : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U✝))
x : ↑(toTopCat Y)
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
hU : (coequalizer.π f.val g.val).base x ∈ U
s : (forget CommRingCat).obj ((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
ha :
IsUnit ((TopCat.Presheaf.germ Y.presheaf { val := x, property := hU }) (((coequalizer.π f.val g.val).c.app (op U)) s))
⊢ IsUnit
((TopCat.Presheaf.germ (coequalizer f.val g.val).toPresheafedSpace.presheaf
{ val := (coequalizer.π f.val g.val).base x, property := hU })
s)
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
|
let V := imageBasicOpen f g U s
|
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.228_0.tE6q65npbp8AX2g
|
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case map_nonunit.intro.intro.intro
X Y : LocallyRingedSpace
f g : X ⟶ Y
U✝ : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s✝ : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U✝))
x : ↑(toTopCat Y)
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
hU : (coequalizer.π f.val g.val).base x ∈ U
s : (forget CommRingCat).obj ((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
ha :
IsUnit ((TopCat.Presheaf.germ Y.presheaf { val := x, property := hU }) (((coequalizer.π f.val g.val).c.app (op U)) s))
V : Opens ↑(toTopCat Y) := imageBasicOpen f g U s
⊢ IsUnit
((TopCat.Presheaf.germ (coequalizer f.val g.val).toPresheafedSpace.presheaf
{ val := (coequalizer.π f.val g.val).base x, property := hU })
s)
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
|
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
|
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.228_0.tE6q65npbp8AX2g
|
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case map_nonunit.intro.intro.intro
X Y : LocallyRingedSpace
f g : X ⟶ Y
U✝ : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s✝ : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U✝))
x : ↑(toTopCat Y)
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
hU : (coequalizer.π f.val g.val).base x ∈ U
s : (forget CommRingCat).obj ((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
ha :
IsUnit ((TopCat.Presheaf.germ Y.presheaf { val := x, property := hU }) (((coequalizer.π f.val g.val).c.app (op U)) s))
V : Opens ↑(toTopCat Y) := imageBasicOpen f g U s
hV : ⇑(coequalizer.π f.val g.val).base ⁻¹' (⇑(coequalizer.π f.val g.val).base '' V.carrier) = V.carrier
⊢ IsUnit
((TopCat.Presheaf.germ (coequalizer f.val g.val).toPresheafedSpace.presheaf
{ val := (coequalizer.π f.val g.val).base x, property := hU })
s)
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
|
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
|
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.228_0.tE6q65npbp8AX2g
|
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case map_nonunit.intro.intro.intro
X Y : LocallyRingedSpace
f g : X ⟶ Y
U✝ : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s✝ : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U✝))
x : ↑(toTopCat Y)
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
hU : (coequalizer.π f.val g.val).base x ∈ U
s : (forget CommRingCat).obj ((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
ha :
IsUnit ((TopCat.Presheaf.germ Y.presheaf { val := x, property := hU }) (((coequalizer.π f.val g.val).c.app (op U)) s))
V : Opens ↑(toTopCat Y) := imageBasicOpen f g U s
hV : ⇑(coequalizer.π f.val g.val).base ⁻¹' (⇑(coequalizer.π f.val g.val).base '' V.carrier) = V.carrier
hV' :
V =
{ carrier := ⇑(coequalizer.π f.val g.val).base ⁻¹' (⇑(coequalizer.π f.val g.val).base '' V.carrier),
is_open' :=
(_ : IsOpen (⇑(coequalizer.π f.val g.val).base ⁻¹' (⇑(coequalizer.π f.val g.val).base '' V.carrier))) }
⊢ IsUnit
((TopCat.Presheaf.germ (coequalizer f.val g.val).toPresheafedSpace.presheaf
{ val := (coequalizer.π f.val g.val).base x, property := hU })
s)
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
|
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
|
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.228_0.tE6q65npbp8AX2g
|
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case map_nonunit.intro.intro.intro
X Y : LocallyRingedSpace
f g : X ⟶ Y
U✝ : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s✝ : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U✝))
x : ↑(toTopCat Y)
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
hU : (coequalizer.π f.val g.val).base x ∈ U
s : (forget CommRingCat).obj ((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
ha :
IsUnit ((TopCat.Presheaf.germ Y.presheaf { val := x, property := hU }) (((coequalizer.π f.val g.val).c.app (op U)) s))
V : Opens ↑(toTopCat Y) := imageBasicOpen f g U s
hV : ⇑(coequalizer.π f.val g.val).base ⁻¹' (⇑(coequalizer.π f.val g.val).base '' V.carrier) = V.carrier
hV' :
V =
{ carrier := ⇑(coequalizer.π f.val g.val).base ⁻¹' (⇑(coequalizer.π f.val g.val).base '' V.carrier),
is_open' :=
(_ : IsOpen (⇑(coequalizer.π f.val g.val).base ⁻¹' (⇑(coequalizer.π f.val g.val).base '' V.carrier))) }
V_open : IsOpen (⇑(coequalizer.π f.val g.val).base '' V.carrier)
⊢ IsUnit
((TopCat.Presheaf.germ (coequalizer f.val g.val).toPresheafedSpace.presheaf
{ val := (coequalizer.π f.val g.val).base x, property := hU })
s)
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
|
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
|
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.228_0.tE6q65npbp8AX2g
|
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case map_nonunit.intro.intro.intro
X Y : LocallyRingedSpace
f g : X ⟶ Y
U✝ : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s✝ : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U✝))
x : ↑(toTopCat Y)
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
hU : (coequalizer.π f.val g.val).base x ∈ U
s : (forget CommRingCat).obj ((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
ha :
IsUnit ((TopCat.Presheaf.germ Y.presheaf { val := x, property := hU }) (((coequalizer.π f.val g.val).c.app (op U)) s))
V : Opens ↑(toTopCat Y) := imageBasicOpen f g U s
hV : ⇑(coequalizer.π f.val g.val).base ⁻¹' (⇑(coequalizer.π f.val g.val).base '' V.carrier) = V.carrier
hV' :
V =
{ carrier := ⇑(coequalizer.π f.val g.val).base ⁻¹' (⇑(coequalizer.π f.val g.val).base '' V.carrier),
is_open' :=
(_ : IsOpen (⇑(coequalizer.π f.val g.val).base ⁻¹' (⇑(coequalizer.π f.val g.val).base '' V.carrier))) }
V_open : IsOpen (⇑(coequalizer.π f.val g.val).base '' V.carrier)
VleU : { carrier := ⇑(coequalizer.π f.val g.val).base '' V.carrier, is_open' := V_open } ≤ U
⊢ IsUnit
((TopCat.Presheaf.germ (coequalizer f.val g.val).toPresheafedSpace.presheaf
{ val := (coequalizer.π f.val g.val).base x, property := hU })
s)
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
|
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
|
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.228_0.tE6q65npbp8AX2g
|
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case map_nonunit.intro.intro.intro
X Y : LocallyRingedSpace
f g : X ⟶ Y
U✝ : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s✝ : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U✝))
x : ↑(toTopCat Y)
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
hU : (coequalizer.π f.val g.val).base x ∈ U
s : (forget CommRingCat).obj ((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
ha :
IsUnit ((TopCat.Presheaf.germ Y.presheaf { val := x, property := hU }) (((coequalizer.π f.val g.val).c.app (op U)) s))
V : Opens ↑(toTopCat Y) := imageBasicOpen f g U s
hV : ⇑(coequalizer.π f.val g.val).base ⁻¹' (⇑(coequalizer.π f.val g.val).base '' V.carrier) = V.carrier
hV' :
V =
{ carrier := ⇑(coequalizer.π f.val g.val).base ⁻¹' (⇑(coequalizer.π f.val g.val).base '' V.carrier),
is_open' :=
(_ : IsOpen (⇑(coequalizer.π f.val g.val).base ⁻¹' (⇑(coequalizer.π f.val g.val).base '' V.carrier))) }
V_open : IsOpen (⇑(coequalizer.π f.val g.val).base '' V.carrier)
VleU : { carrier := ⇑(coequalizer.π f.val g.val).base '' V.carrier, is_open' := V_open } ≤ U
hxV : x ∈ V
⊢ IsUnit
((TopCat.Presheaf.germ (coequalizer f.val g.val).toPresheafedSpace.presheaf
{ val := (coequalizer.π f.val g.val).base x, property := hU })
s)
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
|
erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]
|
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.228_0.tE6q65npbp8AX2g
|
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case map_nonunit.intro.intro.intro
X Y : LocallyRingedSpace
f g : X ⟶ Y
U✝ : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s✝ : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U✝))
x : ↑(toTopCat Y)
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
hU : (coequalizer.π f.val g.val).base x ∈ U
s : (forget CommRingCat).obj ((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
ha :
IsUnit ((TopCat.Presheaf.germ Y.presheaf { val := x, property := hU }) (((coequalizer.π f.val g.val).c.app (op U)) s))
V : Opens ↑(toTopCat Y) := imageBasicOpen f g U s
hV : ⇑(coequalizer.π f.val g.val).base ⁻¹' (⇑(coequalizer.π f.val g.val).base '' V.carrier) = V.carrier
hV' :
V =
{ carrier := ⇑(coequalizer.π f.val g.val).base ⁻¹' (⇑(coequalizer.π f.val g.val).base '' V.carrier),
is_open' :=
(_ : IsOpen (⇑(coequalizer.π f.val g.val).base ⁻¹' (⇑(coequalizer.π f.val g.val).base '' V.carrier))) }
V_open : IsOpen (⇑(coequalizer.π f.val g.val).base '' V.carrier)
VleU : { carrier := ⇑(coequalizer.π f.val g.val).base '' V.carrier, is_open' := V_open } ≤ U
hxV : x ∈ V
⊢ IsUnit
((TopCat.Presheaf.germ (coequalizer f.val g.val).toPresheafedSpace.presheaf
{ val := (coequalizer.π f.val g.val).base x,
property := (_ : (coequalizer.π f.val g.val).base x ∈ ⇑(coequalizer.π f.val g.val).base '' V.carrier) })
(((coequalizer f.val g.val).toPresheafedSpace.presheaf.map (homOfLE VleU).op) s))
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]
|
apply RingHom.isUnit_map
|
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.228_0.tE6q65npbp8AX2g
|
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case map_nonunit.intro.intro.intro.a
X Y : LocallyRingedSpace
f g : X ⟶ Y
U✝ : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s✝ : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U✝))
x : ↑(toTopCat Y)
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
hU : (coequalizer.π f.val g.val).base x ∈ U
s : (forget CommRingCat).obj ((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
ha :
IsUnit ((TopCat.Presheaf.germ Y.presheaf { val := x, property := hU }) (((coequalizer.π f.val g.val).c.app (op U)) s))
V : Opens ↑(toTopCat Y) := imageBasicOpen f g U s
hV : ⇑(coequalizer.π f.val g.val).base ⁻¹' (⇑(coequalizer.π f.val g.val).base '' V.carrier) = V.carrier
hV' :
V =
{ carrier := ⇑(coequalizer.π f.val g.val).base ⁻¹' (⇑(coequalizer.π f.val g.val).base '' V.carrier),
is_open' :=
(_ : IsOpen (⇑(coequalizer.π f.val g.val).base ⁻¹' (⇑(coequalizer.π f.val g.val).base '' V.carrier))) }
V_open : IsOpen (⇑(coequalizer.π f.val g.val).base '' V.carrier)
VleU : { carrier := ⇑(coequalizer.π f.val g.val).base '' V.carrier, is_open' := V_open } ≤ U
hxV : x ∈ V
⊢ IsUnit (((coequalizer f.val g.val).toPresheafedSpace.presheaf.map (homOfLE VleU).op) s)
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]
apply RingHom.isUnit_map
|
rw [← isUnit_map_iff ((coequalizer.π f.val g.val : _).c.app _), ← comp_apply,
NatTrans.naturality, comp_apply, TopCat.Presheaf.pushforwardObj_map, ←
isUnit_map_iff (Y.presheaf.map (eqToHom hV').op)]
|
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]
apply RingHom.isUnit_map
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.228_0.tE6q65npbp8AX2g
|
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case map_nonunit.intro.intro.intro.a
X Y : LocallyRingedSpace
f g : X ⟶ Y
U✝ : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s✝ : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U✝))
x : ↑(toTopCat Y)
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
hU : (coequalizer.π f.val g.val).base x ∈ U
s : (forget CommRingCat).obj ((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
ha :
IsUnit ((TopCat.Presheaf.germ Y.presheaf { val := x, property := hU }) (((coequalizer.π f.val g.val).c.app (op U)) s))
V : Opens ↑(toTopCat Y) := imageBasicOpen f g U s
hV : ⇑(coequalizer.π f.val g.val).base ⁻¹' (⇑(coequalizer.π f.val g.val).base '' V.carrier) = V.carrier
hV' :
V =
{ carrier := ⇑(coequalizer.π f.val g.val).base ⁻¹' (⇑(coequalizer.π f.val g.val).base '' V.carrier),
is_open' :=
(_ : IsOpen (⇑(coequalizer.π f.val g.val).base ⁻¹' (⇑(coequalizer.π f.val g.val).base '' V.carrier))) }
V_open : IsOpen (⇑(coequalizer.π f.val g.val).base '' V.carrier)
VleU : { carrier := ⇑(coequalizer.π f.val g.val).base '' V.carrier, is_open' := V_open } ≤ U
hxV : x ∈ V
⊢ IsUnit
((Y.presheaf.map (eqToHom hV').op)
((Y.presheaf.map ((Opens.map (coequalizer.π f.val g.val).base).op.map (homOfLE VleU).op))
(((coequalizer.π f.val g.val).c.app (op U)) s)))
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]
apply RingHom.isUnit_map
rw [← isUnit_map_iff ((coequalizer.π f.val g.val : _).c.app _), ← comp_apply,
NatTrans.naturality, comp_apply, TopCat.Presheaf.pushforwardObj_map, ←
isUnit_map_iff (Y.presheaf.map (eqToHom hV').op)]
-- Porting note : change `rw` to `erw`
|
erw [← comp_apply, ← comp_apply, ← Y.presheaf.map_comp]
|
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]
apply RingHom.isUnit_map
rw [← isUnit_map_iff ((coequalizer.π f.val g.val : _).c.app _), ← comp_apply,
NatTrans.naturality, comp_apply, TopCat.Presheaf.pushforwardObj_map, ←
isUnit_map_iff (Y.presheaf.map (eqToHom hV').op)]
-- Porting note : change `rw` to `erw`
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.228_0.tE6q65npbp8AX2g
|
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case map_nonunit.intro.intro.intro.a
X Y : LocallyRingedSpace
f g : X ⟶ Y
U✝ : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
s✝ : ↑((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U✝))
x : ↑(toTopCat Y)
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
hU : (coequalizer.π f.val g.val).base x ∈ U
s : (forget CommRingCat).obj ((coequalizer f.val g.val).toPresheafedSpace.presheaf.obj (op U))
ha :
IsUnit ((TopCat.Presheaf.germ Y.presheaf { val := x, property := hU }) (((coequalizer.π f.val g.val).c.app (op U)) s))
V : Opens ↑(toTopCat Y) := imageBasicOpen f g U s
hV : ⇑(coequalizer.π f.val g.val).base ⁻¹' (⇑(coequalizer.π f.val g.val).base '' V.carrier) = V.carrier
hV' :
V =
{ carrier := ⇑(coequalizer.π f.val g.val).base ⁻¹' (⇑(coequalizer.π f.val g.val).base '' V.carrier),
is_open' :=
(_ : IsOpen (⇑(coequalizer.π f.val g.val).base ⁻¹' (⇑(coequalizer.π f.val g.val).base '' V.carrier))) }
V_open : IsOpen (⇑(coequalizer.π f.val g.val).base '' V.carrier)
VleU : { carrier := ⇑(coequalizer.π f.val g.val).base '' V.carrier, is_open' := V_open } ≤ U
hxV : x ∈ V
⊢ IsUnit
(((coequalizer.π f.val g.val).c.app (op U) ≫
Y.presheaf.map ((Opens.map (coequalizer.π f.val g.val).base).op.map (homOfLE VleU).op ≫ (eqToHom hV').op))
s)
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]
apply RingHom.isUnit_map
rw [← isUnit_map_iff ((coequalizer.π f.val g.val : _).c.app _), ← comp_apply,
NatTrans.naturality, comp_apply, TopCat.Presheaf.pushforwardObj_map, ←
isUnit_map_iff (Y.presheaf.map (eqToHom hV').op)]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← comp_apply, ← Y.presheaf.map_comp]
|
convert @RingedSpace.isUnit_res_basicOpen Y.toRingedSpace (unop _)
(((coequalizer.π f.val g.val).c.app (op U)) s)
|
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]
apply RingHom.isUnit_map
rw [← isUnit_map_iff ((coequalizer.π f.val g.val : _).c.app _), ← comp_apply,
NatTrans.naturality, comp_apply, TopCat.Presheaf.pushforwardObj_map, ←
isUnit_map_iff (Y.presheaf.map (eqToHom hV').op)]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← comp_apply, ← Y.presheaf.map_comp]
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.228_0.tE6q65npbp8AX2g
|
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
X Y : LocallyRingedSpace
f g : X ⟶ Y
x : ↑↑(Limits.coequalizer f.val g.val).toPresheafedSpace
⊢ LocalRing ↑(TopCat.Presheaf.stalk (Limits.coequalizer f.val g.val).toPresheafedSpace.presheaf x)
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]
apply RingHom.isUnit_map
rw [← isUnit_map_iff ((coequalizer.π f.val g.val : _).c.app _), ← comp_apply,
NatTrans.naturality, comp_apply, TopCat.Presheaf.pushforwardObj_map, ←
isUnit_map_iff (Y.presheaf.map (eqToHom hV').op)]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← comp_apply, ← Y.presheaf.map_comp]
convert @RingedSpace.isUnit_res_basicOpen Y.toRingedSpace (unop _)
(((coequalizer.π f.val g.val).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_stalk_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_stalk_isLocalRingHom
end HasCoequalizer
/-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. -/
noncomputable def coequalizer : LocallyRingedSpace where
toSheafedSpace := Limits.coequalizer f.1 g.1
localRing x := by
|
obtain ⟨y, rfl⟩ :=
(TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x
|
/-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. -/
noncomputable def coequalizer : LocallyRingedSpace where
toSheafedSpace := Limits.coequalizer f.1 g.1
localRing x := by
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.260_0.tE6q65npbp8AX2g
|
/-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. -/
noncomputable def coequalizer : LocallyRingedSpace where
toSheafedSpace
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case intro
X Y : LocallyRingedSpace
f g : X ⟶ Y
y : (forget TopCat).obj ↑Y.toPresheafedSpace
⊢ LocalRing
↑(TopCat.Presheaf.stalk (Limits.coequalizer f.val g.val).toPresheafedSpace.presheaf
((coequalizer.π f.val g.val).base y))
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]
apply RingHom.isUnit_map
rw [← isUnit_map_iff ((coequalizer.π f.val g.val : _).c.app _), ← comp_apply,
NatTrans.naturality, comp_apply, TopCat.Presheaf.pushforwardObj_map, ←
isUnit_map_iff (Y.presheaf.map (eqToHom hV').op)]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← comp_apply, ← Y.presheaf.map_comp]
convert @RingedSpace.isUnit_res_basicOpen Y.toRingedSpace (unop _)
(((coequalizer.π f.val g.val).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_stalk_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_stalk_isLocalRingHom
end HasCoequalizer
/-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. -/
noncomputable def coequalizer : LocallyRingedSpace where
toSheafedSpace := Limits.coequalizer f.1 g.1
localRing x := by
obtain ⟨y, rfl⟩ :=
(TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x
|
exact (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) y).domain_localRing
|
/-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. -/
noncomputable def coequalizer : LocallyRingedSpace where
toSheafedSpace := Limits.coequalizer f.1 g.1
localRing x := by
obtain ⟨y, rfl⟩ :=
(TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.260_0.tE6q65npbp8AX2g
|
/-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. -/
noncomputable def coequalizer : LocallyRingedSpace where
toSheafedSpace
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
X✝ Y✝ : LocallyRingedSpace
f✝ g✝ : X✝ ⟶ Y✝
X Y : RingedSpace
f g : X ⟶ Y
H : f = g
x : ↑↑X.toPresheafedSpace
h : IsLocalRingHom (PresheafedSpace.stalkMap f x)
⊢ IsLocalRingHom (PresheafedSpace.stalkMap g x)
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]
apply RingHom.isUnit_map
rw [← isUnit_map_iff ((coequalizer.π f.val g.val : _).c.app _), ← comp_apply,
NatTrans.naturality, comp_apply, TopCat.Presheaf.pushforwardObj_map, ←
isUnit_map_iff (Y.presheaf.map (eqToHom hV').op)]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← comp_apply, ← Y.presheaf.map_comp]
convert @RingedSpace.isUnit_res_basicOpen Y.toRingedSpace (unop _)
(((coequalizer.π f.val g.val).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_stalk_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_stalk_isLocalRingHom
end HasCoequalizer
/-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. -/
noncomputable def coequalizer : LocallyRingedSpace where
toSheafedSpace := Limits.coequalizer f.1 g.1
localRing x := by
obtain ⟨y, rfl⟩ :=
(TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x
exact (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) y).domain_localRing
#align algebraic_geometry.LocallyRingedSpace.coequalizer AlgebraicGeometry.LocallyRingedSpace.coequalizer
/-- The explicit coequalizer cofork of locally ringed spaces. -/
noncomputable def coequalizerCofork : Cofork f g :=
@Cofork.ofπ _ _ _ _ f g (coequalizer f g) ⟨coequalizer.π f.1 g.1,
-- Porting note : this used to be automatic
HasCoequalizer.coequalizer_π_stalk_isLocalRingHom _ _⟩
(LocallyRingedSpace.Hom.ext _ _ (coequalizer.condition f.1 g.1))
#align algebraic_geometry.LocallyRingedSpace.coequalizer_cofork AlgebraicGeometry.LocallyRingedSpace.coequalizerCofork
theorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
(h : IsLocalRingHom (PresheafedSpace.stalkMap f x)) :
IsLocalRingHom (PresheafedSpace.stalkMap g x) := by
|
rw [PresheafedSpace.stalkMap.congr_hom _ _ H.symm x]
|
theorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
(h : IsLocalRingHom (PresheafedSpace.stalkMap f x)) :
IsLocalRingHom (PresheafedSpace.stalkMap g x) := by
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.278_0.tE6q65npbp8AX2g
|
theorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
(h : IsLocalRingHom (PresheafedSpace.stalkMap f x)) :
IsLocalRingHom (PresheafedSpace.stalkMap g x)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
X✝ Y✝ : LocallyRingedSpace
f✝ g✝ : X✝ ⟶ Y✝
X Y : RingedSpace
f g : X ⟶ Y
H : f = g
x : ↑↑X.toPresheafedSpace
h : IsLocalRingHom (PresheafedSpace.stalkMap f x)
⊢ IsLocalRingHom
(eqToHom
(_ :
PresheafedSpace.stalk Y.toPresheafedSpace (g.base x) = PresheafedSpace.stalk Y.toPresheafedSpace (f.base x)) ≫
PresheafedSpace.stalkMap f x)
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]
apply RingHom.isUnit_map
rw [← isUnit_map_iff ((coequalizer.π f.val g.val : _).c.app _), ← comp_apply,
NatTrans.naturality, comp_apply, TopCat.Presheaf.pushforwardObj_map, ←
isUnit_map_iff (Y.presheaf.map (eqToHom hV').op)]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← comp_apply, ← Y.presheaf.map_comp]
convert @RingedSpace.isUnit_res_basicOpen Y.toRingedSpace (unop _)
(((coequalizer.π f.val g.val).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_stalk_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_stalk_isLocalRingHom
end HasCoequalizer
/-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. -/
noncomputable def coequalizer : LocallyRingedSpace where
toSheafedSpace := Limits.coequalizer f.1 g.1
localRing x := by
obtain ⟨y, rfl⟩ :=
(TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x
exact (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) y).domain_localRing
#align algebraic_geometry.LocallyRingedSpace.coequalizer AlgebraicGeometry.LocallyRingedSpace.coequalizer
/-- The explicit coequalizer cofork of locally ringed spaces. -/
noncomputable def coequalizerCofork : Cofork f g :=
@Cofork.ofπ _ _ _ _ f g (coequalizer f g) ⟨coequalizer.π f.1 g.1,
-- Porting note : this used to be automatic
HasCoequalizer.coequalizer_π_stalk_isLocalRingHom _ _⟩
(LocallyRingedSpace.Hom.ext _ _ (coequalizer.condition f.1 g.1))
#align algebraic_geometry.LocallyRingedSpace.coequalizer_cofork AlgebraicGeometry.LocallyRingedSpace.coequalizerCofork
theorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
(h : IsLocalRingHom (PresheafedSpace.stalkMap f x)) :
IsLocalRingHom (PresheafedSpace.stalkMap g x) := by
rw [PresheafedSpace.stalkMap.congr_hom _ _ H.symm x];
|
infer_instance
|
theorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
(h : IsLocalRingHom (PresheafedSpace.stalkMap f x)) :
IsLocalRingHom (PresheafedSpace.stalkMap g x) := by
rw [PresheafedSpace.stalkMap.congr_hom _ _ H.symm x];
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.278_0.tE6q65npbp8AX2g
|
theorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
(h : IsLocalRingHom (PresheafedSpace.stalkMap f x)) :
IsLocalRingHom (PresheafedSpace.stalkMap g x)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
X Y : LocallyRingedSpace
f g : X ⟶ Y
⊢ IsColimit (coequalizerCofork f g)
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]
apply RingHom.isUnit_map
rw [← isUnit_map_iff ((coequalizer.π f.val g.val : _).c.app _), ← comp_apply,
NatTrans.naturality, comp_apply, TopCat.Presheaf.pushforwardObj_map, ←
isUnit_map_iff (Y.presheaf.map (eqToHom hV').op)]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← comp_apply, ← Y.presheaf.map_comp]
convert @RingedSpace.isUnit_res_basicOpen Y.toRingedSpace (unop _)
(((coequalizer.π f.val g.val).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_stalk_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_stalk_isLocalRingHom
end HasCoequalizer
/-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. -/
noncomputable def coequalizer : LocallyRingedSpace where
toSheafedSpace := Limits.coequalizer f.1 g.1
localRing x := by
obtain ⟨y, rfl⟩ :=
(TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x
exact (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) y).domain_localRing
#align algebraic_geometry.LocallyRingedSpace.coequalizer AlgebraicGeometry.LocallyRingedSpace.coequalizer
/-- The explicit coequalizer cofork of locally ringed spaces. -/
noncomputable def coequalizerCofork : Cofork f g :=
@Cofork.ofπ _ _ _ _ f g (coequalizer f g) ⟨coequalizer.π f.1 g.1,
-- Porting note : this used to be automatic
HasCoequalizer.coequalizer_π_stalk_isLocalRingHom _ _⟩
(LocallyRingedSpace.Hom.ext _ _ (coequalizer.condition f.1 g.1))
#align algebraic_geometry.LocallyRingedSpace.coequalizer_cofork AlgebraicGeometry.LocallyRingedSpace.coequalizerCofork
theorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
(h : IsLocalRingHom (PresheafedSpace.stalkMap f x)) :
IsLocalRingHom (PresheafedSpace.stalkMap g x) := by
rw [PresheafedSpace.stalkMap.congr_hom _ _ H.symm x]; infer_instance
#align algebraic_geometry.LocallyRingedSpace.is_local_ring_hom_stalk_map_congr AlgebraicGeometry.LocallyRingedSpace.isLocalRingHom_stalkMap_congr
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
|
apply Cofork.IsColimit.mk'
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case create
X Y : LocallyRingedSpace
f g : X ⟶ Y
⊢ (s : Cofork f g) →
{ l //
Cofork.π (coequalizerCofork f g) ≫ l = Cofork.π s ∧
∀
{m :
((Functor.const WalkingParallelPair).obj (coequalizerCofork f g).pt).obj WalkingParallelPair.one ⟶
((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.one},
Cofork.π (coequalizerCofork f g) ≫ m = Cofork.π s → m = l }
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]
apply RingHom.isUnit_map
rw [← isUnit_map_iff ((coequalizer.π f.val g.val : _).c.app _), ← comp_apply,
NatTrans.naturality, comp_apply, TopCat.Presheaf.pushforwardObj_map, ←
isUnit_map_iff (Y.presheaf.map (eqToHom hV').op)]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← comp_apply, ← Y.presheaf.map_comp]
convert @RingedSpace.isUnit_res_basicOpen Y.toRingedSpace (unop _)
(((coequalizer.π f.val g.val).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_stalk_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_stalk_isLocalRingHom
end HasCoequalizer
/-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. -/
noncomputable def coequalizer : LocallyRingedSpace where
toSheafedSpace := Limits.coequalizer f.1 g.1
localRing x := by
obtain ⟨y, rfl⟩ :=
(TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x
exact (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) y).domain_localRing
#align algebraic_geometry.LocallyRingedSpace.coequalizer AlgebraicGeometry.LocallyRingedSpace.coequalizer
/-- The explicit coequalizer cofork of locally ringed spaces. -/
noncomputable def coequalizerCofork : Cofork f g :=
@Cofork.ofπ _ _ _ _ f g (coequalizer f g) ⟨coequalizer.π f.1 g.1,
-- Porting note : this used to be automatic
HasCoequalizer.coequalizer_π_stalk_isLocalRingHom _ _⟩
(LocallyRingedSpace.Hom.ext _ _ (coequalizer.condition f.1 g.1))
#align algebraic_geometry.LocallyRingedSpace.coequalizer_cofork AlgebraicGeometry.LocallyRingedSpace.coequalizerCofork
theorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
(h : IsLocalRingHom (PresheafedSpace.stalkMap f x)) :
IsLocalRingHom (PresheafedSpace.stalkMap g x) := by
rw [PresheafedSpace.stalkMap.congr_hom _ _ H.symm x]; infer_instance
#align algebraic_geometry.LocallyRingedSpace.is_local_ring_hom_stalk_map_congr AlgebraicGeometry.LocallyRingedSpace.isLocalRingHom_stalkMap_congr
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
|
intro s
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case create
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
⊢ { l //
Cofork.π (coequalizerCofork f g) ≫ l = Cofork.π s ∧
∀
{m :
((Functor.const WalkingParallelPair).obj (coequalizerCofork f g).pt).obj WalkingParallelPair.one ⟶
((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.one},
Cofork.π (coequalizerCofork f g) ≫ m = Cofork.π s → m = l }
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]
apply RingHom.isUnit_map
rw [← isUnit_map_iff ((coequalizer.π f.val g.val : _).c.app _), ← comp_apply,
NatTrans.naturality, comp_apply, TopCat.Presheaf.pushforwardObj_map, ←
isUnit_map_iff (Y.presheaf.map (eqToHom hV').op)]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← comp_apply, ← Y.presheaf.map_comp]
convert @RingedSpace.isUnit_res_basicOpen Y.toRingedSpace (unop _)
(((coequalizer.π f.val g.val).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_stalk_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_stalk_isLocalRingHom
end HasCoequalizer
/-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. -/
noncomputable def coequalizer : LocallyRingedSpace where
toSheafedSpace := Limits.coequalizer f.1 g.1
localRing x := by
obtain ⟨y, rfl⟩ :=
(TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x
exact (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) y).domain_localRing
#align algebraic_geometry.LocallyRingedSpace.coequalizer AlgebraicGeometry.LocallyRingedSpace.coequalizer
/-- The explicit coequalizer cofork of locally ringed spaces. -/
noncomputable def coequalizerCofork : Cofork f g :=
@Cofork.ofπ _ _ _ _ f g (coequalizer f g) ⟨coequalizer.π f.1 g.1,
-- Porting note : this used to be automatic
HasCoequalizer.coequalizer_π_stalk_isLocalRingHom _ _⟩
(LocallyRingedSpace.Hom.ext _ _ (coequalizer.condition f.1 g.1))
#align algebraic_geometry.LocallyRingedSpace.coequalizer_cofork AlgebraicGeometry.LocallyRingedSpace.coequalizerCofork
theorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
(h : IsLocalRingHom (PresheafedSpace.stalkMap f x)) :
IsLocalRingHom (PresheafedSpace.stalkMap g x) := by
rw [PresheafedSpace.stalkMap.congr_hom _ _ H.symm x]; infer_instance
#align algebraic_geometry.LocallyRingedSpace.is_local_ring_hom_stalk_map_congr AlgebraicGeometry.LocallyRingedSpace.isLocalRingHom_stalkMap_congr
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
|
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
⊢ f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]
apply RingHom.isUnit_map
rw [← isUnit_map_iff ((coequalizer.π f.val g.val : _).c.app _), ← comp_apply,
NatTrans.naturality, comp_apply, TopCat.Presheaf.pushforwardObj_map, ←
isUnit_map_iff (Y.presheaf.map (eqToHom hV').op)]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← comp_apply, ← Y.presheaf.map_comp]
convert @RingedSpace.isUnit_res_basicOpen Y.toRingedSpace (unop _)
(((coequalizer.π f.val g.val).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_stalk_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_stalk_isLocalRingHom
end HasCoequalizer
/-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. -/
noncomputable def coequalizer : LocallyRingedSpace where
toSheafedSpace := Limits.coequalizer f.1 g.1
localRing x := by
obtain ⟨y, rfl⟩ :=
(TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x
exact (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) y).domain_localRing
#align algebraic_geometry.LocallyRingedSpace.coequalizer AlgebraicGeometry.LocallyRingedSpace.coequalizer
/-- The explicit coequalizer cofork of locally ringed spaces. -/
noncomputable def coequalizerCofork : Cofork f g :=
@Cofork.ofπ _ _ _ _ f g (coequalizer f g) ⟨coequalizer.π f.1 g.1,
-- Porting note : this used to be automatic
HasCoequalizer.coequalizer_π_stalk_isLocalRingHom _ _⟩
(LocallyRingedSpace.Hom.ext _ _ (coequalizer.condition f.1 g.1))
#align algebraic_geometry.LocallyRingedSpace.coequalizer_cofork AlgebraicGeometry.LocallyRingedSpace.coequalizerCofork
theorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
(h : IsLocalRingHom (PresheafedSpace.stalkMap f x)) :
IsLocalRingHom (PresheafedSpace.stalkMap g x) := by
rw [PresheafedSpace.stalkMap.congr_hom _ _ H.symm x]; infer_instance
#align algebraic_geometry.LocallyRingedSpace.is_local_ring_hom_stalk_map_congr AlgebraicGeometry.LocallyRingedSpace.isLocalRingHom_stalkMap_congr
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by
|
injection s.condition
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case create
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
⊢ { l //
Cofork.π (coequalizerCofork f g) ≫ l = Cofork.π s ∧
∀
{m :
((Functor.const WalkingParallelPair).obj (coequalizerCofork f g).pt).obj WalkingParallelPair.one ⟶
((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.one},
Cofork.π (coequalizerCofork f g) ≫ m = Cofork.π s → m = l }
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]
apply RingHom.isUnit_map
rw [← isUnit_map_iff ((coequalizer.π f.val g.val : _).c.app _), ← comp_apply,
NatTrans.naturality, comp_apply, TopCat.Presheaf.pushforwardObj_map, ←
isUnit_map_iff (Y.presheaf.map (eqToHom hV').op)]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← comp_apply, ← Y.presheaf.map_comp]
convert @RingedSpace.isUnit_res_basicOpen Y.toRingedSpace (unop _)
(((coequalizer.π f.val g.val).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_stalk_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_stalk_isLocalRingHom
end HasCoequalizer
/-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. -/
noncomputable def coequalizer : LocallyRingedSpace where
toSheafedSpace := Limits.coequalizer f.1 g.1
localRing x := by
obtain ⟨y, rfl⟩ :=
(TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x
exact (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) y).domain_localRing
#align algebraic_geometry.LocallyRingedSpace.coequalizer AlgebraicGeometry.LocallyRingedSpace.coequalizer
/-- The explicit coequalizer cofork of locally ringed spaces. -/
noncomputable def coequalizerCofork : Cofork f g :=
@Cofork.ofπ _ _ _ _ f g (coequalizer f g) ⟨coequalizer.π f.1 g.1,
-- Porting note : this used to be automatic
HasCoequalizer.coequalizer_π_stalk_isLocalRingHom _ _⟩
(LocallyRingedSpace.Hom.ext _ _ (coequalizer.condition f.1 g.1))
#align algebraic_geometry.LocallyRingedSpace.coequalizer_cofork AlgebraicGeometry.LocallyRingedSpace.coequalizerCofork
theorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
(h : IsLocalRingHom (PresheafedSpace.stalkMap f x)) :
IsLocalRingHom (PresheafedSpace.stalkMap g x) := by
rw [PresheafedSpace.stalkMap.congr_hom _ _ H.symm x]; infer_instance
#align algebraic_geometry.LocallyRingedSpace.is_local_ring_hom_stalk_map_congr AlgebraicGeometry.LocallyRingedSpace.isLocalRingHom_stalkMap_congr
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
|
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case create.refine_1
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
⊢ ∀ (x : ↑↑(coequalizerCofork f g).pt.toPresheafedSpace),
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.desc (Cofork.π s).val e) x)
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]
apply RingHom.isUnit_map
rw [← isUnit_map_iff ((coequalizer.π f.val g.val : _).c.app _), ← comp_apply,
NatTrans.naturality, comp_apply, TopCat.Presheaf.pushforwardObj_map, ←
isUnit_map_iff (Y.presheaf.map (eqToHom hV').op)]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← comp_apply, ← Y.presheaf.map_comp]
convert @RingedSpace.isUnit_res_basicOpen Y.toRingedSpace (unop _)
(((coequalizer.π f.val g.val).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_stalk_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_stalk_isLocalRingHom
end HasCoequalizer
/-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. -/
noncomputable def coequalizer : LocallyRingedSpace where
toSheafedSpace := Limits.coequalizer f.1 g.1
localRing x := by
obtain ⟨y, rfl⟩ :=
(TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x
exact (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) y).domain_localRing
#align algebraic_geometry.LocallyRingedSpace.coequalizer AlgebraicGeometry.LocallyRingedSpace.coequalizer
/-- The explicit coequalizer cofork of locally ringed spaces. -/
noncomputable def coequalizerCofork : Cofork f g :=
@Cofork.ofπ _ _ _ _ f g (coequalizer f g) ⟨coequalizer.π f.1 g.1,
-- Porting note : this used to be automatic
HasCoequalizer.coequalizer_π_stalk_isLocalRingHom _ _⟩
(LocallyRingedSpace.Hom.ext _ _ (coequalizer.condition f.1 g.1))
#align algebraic_geometry.LocallyRingedSpace.coequalizer_cofork AlgebraicGeometry.LocallyRingedSpace.coequalizerCofork
theorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
(h : IsLocalRingHom (PresheafedSpace.stalkMap f x)) :
IsLocalRingHom (PresheafedSpace.stalkMap g x) := by
rw [PresheafedSpace.stalkMap.congr_hom _ _ H.symm x]; infer_instance
#align algebraic_geometry.LocallyRingedSpace.is_local_ring_hom_stalk_map_congr AlgebraicGeometry.LocallyRingedSpace.isLocalRingHom_stalkMap_congr
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
·
|
intro x
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
·
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case create.refine_1
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
x : ↑↑(coequalizerCofork f g).pt.toPresheafedSpace
⊢ IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.desc (Cofork.π s).val e) x)
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]
apply RingHom.isUnit_map
rw [← isUnit_map_iff ((coequalizer.π f.val g.val : _).c.app _), ← comp_apply,
NatTrans.naturality, comp_apply, TopCat.Presheaf.pushforwardObj_map, ←
isUnit_map_iff (Y.presheaf.map (eqToHom hV').op)]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← comp_apply, ← Y.presheaf.map_comp]
convert @RingedSpace.isUnit_res_basicOpen Y.toRingedSpace (unop _)
(((coequalizer.π f.val g.val).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_stalk_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_stalk_isLocalRingHom
end HasCoequalizer
/-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. -/
noncomputable def coequalizer : LocallyRingedSpace where
toSheafedSpace := Limits.coequalizer f.1 g.1
localRing x := by
obtain ⟨y, rfl⟩ :=
(TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x
exact (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) y).domain_localRing
#align algebraic_geometry.LocallyRingedSpace.coequalizer AlgebraicGeometry.LocallyRingedSpace.coequalizer
/-- The explicit coequalizer cofork of locally ringed spaces. -/
noncomputable def coequalizerCofork : Cofork f g :=
@Cofork.ofπ _ _ _ _ f g (coequalizer f g) ⟨coequalizer.π f.1 g.1,
-- Porting note : this used to be automatic
HasCoequalizer.coequalizer_π_stalk_isLocalRingHom _ _⟩
(LocallyRingedSpace.Hom.ext _ _ (coequalizer.condition f.1 g.1))
#align algebraic_geometry.LocallyRingedSpace.coequalizer_cofork AlgebraicGeometry.LocallyRingedSpace.coequalizerCofork
theorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
(h : IsLocalRingHom (PresheafedSpace.stalkMap f x)) :
IsLocalRingHom (PresheafedSpace.stalkMap g x) := by
rw [PresheafedSpace.stalkMap.congr_hom _ _ H.symm x]; infer_instance
#align algebraic_geometry.LocallyRingedSpace.is_local_ring_hom_stalk_map_congr AlgebraicGeometry.LocallyRingedSpace.isLocalRingHom_stalkMap_congr
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
|
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case create.refine_1.intro
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
y : (forget TopCat).obj ↑Y.toPresheafedSpace
⊢ IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.desc (Cofork.π s).val e) ((coequalizer.π f.val g.val).base y))
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]
apply RingHom.isUnit_map
rw [← isUnit_map_iff ((coequalizer.π f.val g.val : _).c.app _), ← comp_apply,
NatTrans.naturality, comp_apply, TopCat.Presheaf.pushforwardObj_map, ←
isUnit_map_iff (Y.presheaf.map (eqToHom hV').op)]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← comp_apply, ← Y.presheaf.map_comp]
convert @RingedSpace.isUnit_res_basicOpen Y.toRingedSpace (unop _)
(((coequalizer.π f.val g.val).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_stalk_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_stalk_isLocalRingHom
end HasCoequalizer
/-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. -/
noncomputable def coequalizer : LocallyRingedSpace where
toSheafedSpace := Limits.coequalizer f.1 g.1
localRing x := by
obtain ⟨y, rfl⟩ :=
(TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x
exact (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) y).domain_localRing
#align algebraic_geometry.LocallyRingedSpace.coequalizer AlgebraicGeometry.LocallyRingedSpace.coequalizer
/-- The explicit coequalizer cofork of locally ringed spaces. -/
noncomputable def coequalizerCofork : Cofork f g :=
@Cofork.ofπ _ _ _ _ f g (coequalizer f g) ⟨coequalizer.π f.1 g.1,
-- Porting note : this used to be automatic
HasCoequalizer.coequalizer_π_stalk_isLocalRingHom _ _⟩
(LocallyRingedSpace.Hom.ext _ _ (coequalizer.condition f.1 g.1))
#align algebraic_geometry.LocallyRingedSpace.coequalizer_cofork AlgebraicGeometry.LocallyRingedSpace.coequalizerCofork
theorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
(h : IsLocalRingHom (PresheafedSpace.stalkMap f x)) :
IsLocalRingHom (PresheafedSpace.stalkMap g x) := by
rw [PresheafedSpace.stalkMap.congr_hom _ _ H.symm x]; infer_instance
#align algebraic_geometry.LocallyRingedSpace.is_local_ring_hom_stalk_map_congr AlgebraicGeometry.LocallyRingedSpace.isLocalRingHom_stalkMap_congr
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
|
set h := _
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case create.refine_1.intro
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
y : (forget TopCat).obj ↑Y.toPresheafedSpace
h : ?m.199155 := ?m.199156
⊢ IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.desc (Cofork.π s).val e) ((coequalizer.π f.val g.val).base y))
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]
apply RingHom.isUnit_map
rw [← isUnit_map_iff ((coequalizer.π f.val g.val : _).c.app _), ← comp_apply,
NatTrans.naturality, comp_apply, TopCat.Presheaf.pushforwardObj_map, ←
isUnit_map_iff (Y.presheaf.map (eqToHom hV').op)]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← comp_apply, ← Y.presheaf.map_comp]
convert @RingedSpace.isUnit_res_basicOpen Y.toRingedSpace (unop _)
(((coequalizer.π f.val g.val).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_stalk_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_stalk_isLocalRingHom
end HasCoequalizer
/-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. -/
noncomputable def coequalizer : LocallyRingedSpace where
toSheafedSpace := Limits.coequalizer f.1 g.1
localRing x := by
obtain ⟨y, rfl⟩ :=
(TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x
exact (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) y).domain_localRing
#align algebraic_geometry.LocallyRingedSpace.coequalizer AlgebraicGeometry.LocallyRingedSpace.coequalizer
/-- The explicit coequalizer cofork of locally ringed spaces. -/
noncomputable def coequalizerCofork : Cofork f g :=
@Cofork.ofπ _ _ _ _ f g (coequalizer f g) ⟨coequalizer.π f.1 g.1,
-- Porting note : this used to be automatic
HasCoequalizer.coequalizer_π_stalk_isLocalRingHom _ _⟩
(LocallyRingedSpace.Hom.ext _ _ (coequalizer.condition f.1 g.1))
#align algebraic_geometry.LocallyRingedSpace.coequalizer_cofork AlgebraicGeometry.LocallyRingedSpace.coequalizerCofork
theorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
(h : IsLocalRingHom (PresheafedSpace.stalkMap f x)) :
IsLocalRingHom (PresheafedSpace.stalkMap g x) := by
rw [PresheafedSpace.stalkMap.congr_hom _ _ H.symm x]; infer_instance
#align algebraic_geometry.LocallyRingedSpace.is_local_ring_hom_stalk_map_congr AlgebraicGeometry.LocallyRingedSpace.isLocalRingHom_stalkMap_congr
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
set h := _
|
change IsLocalRingHom h
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
set h := _
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case create.refine_1.intro
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
y : (forget TopCat).obj ↑Y.toPresheafedSpace
h : ↑(PresheafedSpace.stalk s.pt.toPresheafedSpace
((coequalizer.desc (Cofork.π s).val e).base ((coequalizer.π f.val g.val).base y))) →+*
↑(PresheafedSpace.stalk (coequalizerCofork f g).pt.toPresheafedSpace ((coequalizer.π f.val g.val).base y)) :=
PresheafedSpace.stalkMap (coequalizer.desc (Cofork.π s).val e) ((coequalizer.π f.val g.val).base y)
⊢ IsLocalRingHom h
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]
apply RingHom.isUnit_map
rw [← isUnit_map_iff ((coequalizer.π f.val g.val : _).c.app _), ← comp_apply,
NatTrans.naturality, comp_apply, TopCat.Presheaf.pushforwardObj_map, ←
isUnit_map_iff (Y.presheaf.map (eqToHom hV').op)]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← comp_apply, ← Y.presheaf.map_comp]
convert @RingedSpace.isUnit_res_basicOpen Y.toRingedSpace (unop _)
(((coequalizer.π f.val g.val).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_stalk_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_stalk_isLocalRingHom
end HasCoequalizer
/-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. -/
noncomputable def coequalizer : LocallyRingedSpace where
toSheafedSpace := Limits.coequalizer f.1 g.1
localRing x := by
obtain ⟨y, rfl⟩ :=
(TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x
exact (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) y).domain_localRing
#align algebraic_geometry.LocallyRingedSpace.coequalizer AlgebraicGeometry.LocallyRingedSpace.coequalizer
/-- The explicit coequalizer cofork of locally ringed spaces. -/
noncomputable def coequalizerCofork : Cofork f g :=
@Cofork.ofπ _ _ _ _ f g (coequalizer f g) ⟨coequalizer.π f.1 g.1,
-- Porting note : this used to be automatic
HasCoequalizer.coequalizer_π_stalk_isLocalRingHom _ _⟩
(LocallyRingedSpace.Hom.ext _ _ (coequalizer.condition f.1 g.1))
#align algebraic_geometry.LocallyRingedSpace.coequalizer_cofork AlgebraicGeometry.LocallyRingedSpace.coequalizerCofork
theorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
(h : IsLocalRingHom (PresheafedSpace.stalkMap f x)) :
IsLocalRingHom (PresheafedSpace.stalkMap g x) := by
rw [PresheafedSpace.stalkMap.congr_hom _ _ H.symm x]; infer_instance
#align algebraic_geometry.LocallyRingedSpace.is_local_ring_hom_stalk_map_congr AlgebraicGeometry.LocallyRingedSpace.isLocalRingHom_stalkMap_congr
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
set h := _
change IsLocalRingHom h
|
suffices : IsLocalRingHom ((PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _).comp h)
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
set h := _
change IsLocalRingHom h
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case create.refine_1.intro
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
y : (forget TopCat).obj ↑Y.toPresheafedSpace
h : ↑(PresheafedSpace.stalk s.pt.toPresheafedSpace
((coequalizer.desc (Cofork.π s).val e).base ((coequalizer.π f.val g.val).base y))) →+*
↑(PresheafedSpace.stalk (coequalizerCofork f g).pt.toPresheafedSpace ((coequalizer.π f.val g.val).base y)) :=
PresheafedSpace.stalkMap (coequalizer.desc (Cofork.π s).val e) ((coequalizer.π f.val g.val).base y)
this : IsLocalRingHom (RingHom.comp (PresheafedSpace.stalkMap (Cofork.π (coequalizerCofork f g)).val y) h)
⊢ IsLocalRingHom h
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]
apply RingHom.isUnit_map
rw [← isUnit_map_iff ((coequalizer.π f.val g.val : _).c.app _), ← comp_apply,
NatTrans.naturality, comp_apply, TopCat.Presheaf.pushforwardObj_map, ←
isUnit_map_iff (Y.presheaf.map (eqToHom hV').op)]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← comp_apply, ← Y.presheaf.map_comp]
convert @RingedSpace.isUnit_res_basicOpen Y.toRingedSpace (unop _)
(((coequalizer.π f.val g.val).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_stalk_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_stalk_isLocalRingHom
end HasCoequalizer
/-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. -/
noncomputable def coequalizer : LocallyRingedSpace where
toSheafedSpace := Limits.coequalizer f.1 g.1
localRing x := by
obtain ⟨y, rfl⟩ :=
(TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x
exact (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) y).domain_localRing
#align algebraic_geometry.LocallyRingedSpace.coequalizer AlgebraicGeometry.LocallyRingedSpace.coequalizer
/-- The explicit coequalizer cofork of locally ringed spaces. -/
noncomputable def coequalizerCofork : Cofork f g :=
@Cofork.ofπ _ _ _ _ f g (coequalizer f g) ⟨coequalizer.π f.1 g.1,
-- Porting note : this used to be automatic
HasCoequalizer.coequalizer_π_stalk_isLocalRingHom _ _⟩
(LocallyRingedSpace.Hom.ext _ _ (coequalizer.condition f.1 g.1))
#align algebraic_geometry.LocallyRingedSpace.coequalizer_cofork AlgebraicGeometry.LocallyRingedSpace.coequalizerCofork
theorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
(h : IsLocalRingHom (PresheafedSpace.stalkMap f x)) :
IsLocalRingHom (PresheafedSpace.stalkMap g x) := by
rw [PresheafedSpace.stalkMap.congr_hom _ _ H.symm x]; infer_instance
#align algebraic_geometry.LocallyRingedSpace.is_local_ring_hom_stalk_map_congr AlgebraicGeometry.LocallyRingedSpace.isLocalRingHom_stalkMap_congr
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
set h := _
change IsLocalRingHom h
suffices : IsLocalRingHom ((PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _).comp h)
·
|
apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _)
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
set h := _
change IsLocalRingHom h
suffices : IsLocalRingHom ((PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _).comp h)
·
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case this
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
y : (forget TopCat).obj ↑Y.toPresheafedSpace
h : ↑(PresheafedSpace.stalk s.pt.toPresheafedSpace
((coequalizer.desc (Cofork.π s).val e).base ((coequalizer.π f.val g.val).base y))) →+*
↑(PresheafedSpace.stalk (coequalizerCofork f g).pt.toPresheafedSpace ((coequalizer.π f.val g.val).base y)) :=
PresheafedSpace.stalkMap (coequalizer.desc (Cofork.π s).val e) ((coequalizer.π f.val g.val).base y)
⊢ IsLocalRingHom (RingHom.comp (PresheafedSpace.stalkMap (Cofork.π (coequalizerCofork f g)).val y) h)
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]
apply RingHom.isUnit_map
rw [← isUnit_map_iff ((coequalizer.π f.val g.val : _).c.app _), ← comp_apply,
NatTrans.naturality, comp_apply, TopCat.Presheaf.pushforwardObj_map, ←
isUnit_map_iff (Y.presheaf.map (eqToHom hV').op)]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← comp_apply, ← Y.presheaf.map_comp]
convert @RingedSpace.isUnit_res_basicOpen Y.toRingedSpace (unop _)
(((coequalizer.π f.val g.val).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_stalk_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_stalk_isLocalRingHom
end HasCoequalizer
/-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. -/
noncomputable def coequalizer : LocallyRingedSpace where
toSheafedSpace := Limits.coequalizer f.1 g.1
localRing x := by
obtain ⟨y, rfl⟩ :=
(TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x
exact (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) y).domain_localRing
#align algebraic_geometry.LocallyRingedSpace.coequalizer AlgebraicGeometry.LocallyRingedSpace.coequalizer
/-- The explicit coequalizer cofork of locally ringed spaces. -/
noncomputable def coequalizerCofork : Cofork f g :=
@Cofork.ofπ _ _ _ _ f g (coequalizer f g) ⟨coequalizer.π f.1 g.1,
-- Porting note : this used to be automatic
HasCoequalizer.coequalizer_π_stalk_isLocalRingHom _ _⟩
(LocallyRingedSpace.Hom.ext _ _ (coequalizer.condition f.1 g.1))
#align algebraic_geometry.LocallyRingedSpace.coequalizer_cofork AlgebraicGeometry.LocallyRingedSpace.coequalizerCofork
theorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
(h : IsLocalRingHom (PresheafedSpace.stalkMap f x)) :
IsLocalRingHom (PresheafedSpace.stalkMap g x) := by
rw [PresheafedSpace.stalkMap.congr_hom _ _ H.symm x]; infer_instance
#align algebraic_geometry.LocallyRingedSpace.is_local_ring_hom_stalk_map_congr AlgebraicGeometry.LocallyRingedSpace.isLocalRingHom_stalkMap_congr
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
set h := _
change IsLocalRingHom h
suffices : IsLocalRingHom ((PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _).comp h)
· apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _)
|
change IsLocalRingHom (_ ≫ PresheafedSpace.stalkMap (coequalizerCofork f g).π.val y)
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
set h := _
change IsLocalRingHom h
suffices : IsLocalRingHom ((PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _).comp h)
· apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _)
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case this
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
y : (forget TopCat).obj ↑Y.toPresheafedSpace
h : ↑(PresheafedSpace.stalk s.pt.toPresheafedSpace
((coequalizer.desc (Cofork.π s).val e).base ((coequalizer.π f.val g.val).base y))) →+*
↑(PresheafedSpace.stalk (coequalizerCofork f g).pt.toPresheafedSpace ((coequalizer.π f.val g.val).base y)) :=
PresheafedSpace.stalkMap (coequalizer.desc (Cofork.π s).val e) ((coequalizer.π f.val g.val).base y)
⊢ IsLocalRingHom (h ≫ PresheafedSpace.stalkMap (Cofork.π (coequalizerCofork f g)).val y)
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]
apply RingHom.isUnit_map
rw [← isUnit_map_iff ((coequalizer.π f.val g.val : _).c.app _), ← comp_apply,
NatTrans.naturality, comp_apply, TopCat.Presheaf.pushforwardObj_map, ←
isUnit_map_iff (Y.presheaf.map (eqToHom hV').op)]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← comp_apply, ← Y.presheaf.map_comp]
convert @RingedSpace.isUnit_res_basicOpen Y.toRingedSpace (unop _)
(((coequalizer.π f.val g.val).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_stalk_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_stalk_isLocalRingHom
end HasCoequalizer
/-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. -/
noncomputable def coequalizer : LocallyRingedSpace where
toSheafedSpace := Limits.coequalizer f.1 g.1
localRing x := by
obtain ⟨y, rfl⟩ :=
(TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x
exact (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) y).domain_localRing
#align algebraic_geometry.LocallyRingedSpace.coequalizer AlgebraicGeometry.LocallyRingedSpace.coequalizer
/-- The explicit coequalizer cofork of locally ringed spaces. -/
noncomputable def coequalizerCofork : Cofork f g :=
@Cofork.ofπ _ _ _ _ f g (coequalizer f g) ⟨coequalizer.π f.1 g.1,
-- Porting note : this used to be automatic
HasCoequalizer.coequalizer_π_stalk_isLocalRingHom _ _⟩
(LocallyRingedSpace.Hom.ext _ _ (coequalizer.condition f.1 g.1))
#align algebraic_geometry.LocallyRingedSpace.coequalizer_cofork AlgebraicGeometry.LocallyRingedSpace.coequalizerCofork
theorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
(h : IsLocalRingHom (PresheafedSpace.stalkMap f x)) :
IsLocalRingHom (PresheafedSpace.stalkMap g x) := by
rw [PresheafedSpace.stalkMap.congr_hom _ _ H.symm x]; infer_instance
#align algebraic_geometry.LocallyRingedSpace.is_local_ring_hom_stalk_map_congr AlgebraicGeometry.LocallyRingedSpace.isLocalRingHom_stalkMap_congr
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
set h := _
change IsLocalRingHom h
suffices : IsLocalRingHom ((PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _).comp h)
· apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _)
change IsLocalRingHom (_ ≫ PresheafedSpace.stalkMap (coequalizerCofork f g).π.val y)
|
erw [← PresheafedSpace.stalkMap.comp]
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
set h := _
change IsLocalRingHom h
suffices : IsLocalRingHom ((PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _).comp h)
· apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _)
change IsLocalRingHom (_ ≫ PresheafedSpace.stalkMap (coequalizerCofork f g).π.val y)
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case this
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
y : (forget TopCat).obj ↑Y.toPresheafedSpace
h : ↑(PresheafedSpace.stalk s.pt.toPresheafedSpace
((coequalizer.desc (Cofork.π s).val e).base ((coequalizer.π f.val g.val).base y))) →+*
↑(PresheafedSpace.stalk (coequalizerCofork f g).pt.toPresheafedSpace ((coequalizer.π f.val g.val).base y)) :=
PresheafedSpace.stalkMap (coequalizer.desc (Cofork.π s).val e) ((coequalizer.π f.val g.val).base y)
⊢ IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val ≫ coequalizer.desc (Cofork.π s).val e) y)
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]
apply RingHom.isUnit_map
rw [← isUnit_map_iff ((coequalizer.π f.val g.val : _).c.app _), ← comp_apply,
NatTrans.naturality, comp_apply, TopCat.Presheaf.pushforwardObj_map, ←
isUnit_map_iff (Y.presheaf.map (eqToHom hV').op)]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← comp_apply, ← Y.presheaf.map_comp]
convert @RingedSpace.isUnit_res_basicOpen Y.toRingedSpace (unop _)
(((coequalizer.π f.val g.val).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_stalk_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_stalk_isLocalRingHom
end HasCoequalizer
/-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. -/
noncomputable def coequalizer : LocallyRingedSpace where
toSheafedSpace := Limits.coequalizer f.1 g.1
localRing x := by
obtain ⟨y, rfl⟩ :=
(TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x
exact (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) y).domain_localRing
#align algebraic_geometry.LocallyRingedSpace.coequalizer AlgebraicGeometry.LocallyRingedSpace.coequalizer
/-- The explicit coequalizer cofork of locally ringed spaces. -/
noncomputable def coequalizerCofork : Cofork f g :=
@Cofork.ofπ _ _ _ _ f g (coequalizer f g) ⟨coequalizer.π f.1 g.1,
-- Porting note : this used to be automatic
HasCoequalizer.coequalizer_π_stalk_isLocalRingHom _ _⟩
(LocallyRingedSpace.Hom.ext _ _ (coequalizer.condition f.1 g.1))
#align algebraic_geometry.LocallyRingedSpace.coequalizer_cofork AlgebraicGeometry.LocallyRingedSpace.coequalizerCofork
theorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
(h : IsLocalRingHom (PresheafedSpace.stalkMap f x)) :
IsLocalRingHom (PresheafedSpace.stalkMap g x) := by
rw [PresheafedSpace.stalkMap.congr_hom _ _ H.symm x]; infer_instance
#align algebraic_geometry.LocallyRingedSpace.is_local_ring_hom_stalk_map_congr AlgebraicGeometry.LocallyRingedSpace.isLocalRingHom_stalkMap_congr
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
set h := _
change IsLocalRingHom h
suffices : IsLocalRingHom ((PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _).comp h)
· apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _)
change IsLocalRingHom (_ ≫ PresheafedSpace.stalkMap (coequalizerCofork f g).π.val y)
erw [← PresheafedSpace.stalkMap.comp]
|
apply isLocalRingHom_stalkMap_congr _ _ (coequalizer.π_desc s.π.1 e).symm y
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
set h := _
change IsLocalRingHom h
suffices : IsLocalRingHom ((PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _).comp h)
· apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _)
change IsLocalRingHom (_ ≫ PresheafedSpace.stalkMap (coequalizerCofork f g).π.val y)
erw [← PresheafedSpace.stalkMap.comp]
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case this
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
y : (forget TopCat).obj ↑Y.toPresheafedSpace
h : ↑(PresheafedSpace.stalk s.pt.toPresheafedSpace
((coequalizer.desc (Cofork.π s).val e).base ((coequalizer.π f.val g.val).base y))) →+*
↑(PresheafedSpace.stalk (coequalizerCofork f g).pt.toPresheafedSpace ((coequalizer.π f.val g.val).base y)) :=
PresheafedSpace.stalkMap (coequalizer.desc (Cofork.π s).val e) ((coequalizer.π f.val g.val).base y)
⊢ IsLocalRingHom (PresheafedSpace.stalkMap (Cofork.π s).val y)
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]
apply RingHom.isUnit_map
rw [← isUnit_map_iff ((coequalizer.π f.val g.val : _).c.app _), ← comp_apply,
NatTrans.naturality, comp_apply, TopCat.Presheaf.pushforwardObj_map, ←
isUnit_map_iff (Y.presheaf.map (eqToHom hV').op)]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← comp_apply, ← Y.presheaf.map_comp]
convert @RingedSpace.isUnit_res_basicOpen Y.toRingedSpace (unop _)
(((coequalizer.π f.val g.val).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_stalk_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_stalk_isLocalRingHom
end HasCoequalizer
/-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. -/
noncomputable def coequalizer : LocallyRingedSpace where
toSheafedSpace := Limits.coequalizer f.1 g.1
localRing x := by
obtain ⟨y, rfl⟩ :=
(TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x
exact (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) y).domain_localRing
#align algebraic_geometry.LocallyRingedSpace.coequalizer AlgebraicGeometry.LocallyRingedSpace.coequalizer
/-- The explicit coequalizer cofork of locally ringed spaces. -/
noncomputable def coequalizerCofork : Cofork f g :=
@Cofork.ofπ _ _ _ _ f g (coequalizer f g) ⟨coequalizer.π f.1 g.1,
-- Porting note : this used to be automatic
HasCoequalizer.coequalizer_π_stalk_isLocalRingHom _ _⟩
(LocallyRingedSpace.Hom.ext _ _ (coequalizer.condition f.1 g.1))
#align algebraic_geometry.LocallyRingedSpace.coequalizer_cofork AlgebraicGeometry.LocallyRingedSpace.coequalizerCofork
theorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
(h : IsLocalRingHom (PresheafedSpace.stalkMap f x)) :
IsLocalRingHom (PresheafedSpace.stalkMap g x) := by
rw [PresheafedSpace.stalkMap.congr_hom _ _ H.symm x]; infer_instance
#align algebraic_geometry.LocallyRingedSpace.is_local_ring_hom_stalk_map_congr AlgebraicGeometry.LocallyRingedSpace.isLocalRingHom_stalkMap_congr
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
set h := _
change IsLocalRingHom h
suffices : IsLocalRingHom ((PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _).comp h)
· apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _)
change IsLocalRingHom (_ ≫ PresheafedSpace.stalkMap (coequalizerCofork f g).π.val y)
erw [← PresheafedSpace.stalkMap.comp]
apply isLocalRingHom_stalkMap_congr _ _ (coequalizer.π_desc s.π.1 e).symm y
|
infer_instance
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
set h := _
change IsLocalRingHom h
suffices : IsLocalRingHom ((PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _).comp h)
· apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _)
change IsLocalRingHom (_ ≫ PresheafedSpace.stalkMap (coequalizerCofork f g).π.val y)
erw [← PresheafedSpace.stalkMap.comp]
apply isLocalRingHom_stalkMap_congr _ _ (coequalizer.π_desc s.π.1 e).symm y
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case create.refine_2
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
⊢ Cofork.π (coequalizerCofork f g) ≫
{ val := coequalizer.desc (Cofork.π s).val e,
prop :=
(_ :
∀ (x : ↑↑(coequalizerCofork f g).pt.toPresheafedSpace),
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.desc (Cofork.π s).val e) x)) } =
Cofork.π s ∧
∀
{m :
((Functor.const WalkingParallelPair).obj (coequalizerCofork f g).pt).obj WalkingParallelPair.one ⟶
((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.one},
Cofork.π (coequalizerCofork f g) ≫ m = Cofork.π s →
m =
{ val := coequalizer.desc (Cofork.π s).val e,
prop :=
(_ :
∀ (x : ↑↑(coequalizerCofork f g).pt.toPresheafedSpace),
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.desc (Cofork.π s).val e) x)) }
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]
apply RingHom.isUnit_map
rw [← isUnit_map_iff ((coequalizer.π f.val g.val : _).c.app _), ← comp_apply,
NatTrans.naturality, comp_apply, TopCat.Presheaf.pushforwardObj_map, ←
isUnit_map_iff (Y.presheaf.map (eqToHom hV').op)]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← comp_apply, ← Y.presheaf.map_comp]
convert @RingedSpace.isUnit_res_basicOpen Y.toRingedSpace (unop _)
(((coequalizer.π f.val g.val).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_stalk_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_stalk_isLocalRingHom
end HasCoequalizer
/-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. -/
noncomputable def coequalizer : LocallyRingedSpace where
toSheafedSpace := Limits.coequalizer f.1 g.1
localRing x := by
obtain ⟨y, rfl⟩ :=
(TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x
exact (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) y).domain_localRing
#align algebraic_geometry.LocallyRingedSpace.coequalizer AlgebraicGeometry.LocallyRingedSpace.coequalizer
/-- The explicit coequalizer cofork of locally ringed spaces. -/
noncomputable def coequalizerCofork : Cofork f g :=
@Cofork.ofπ _ _ _ _ f g (coequalizer f g) ⟨coequalizer.π f.1 g.1,
-- Porting note : this used to be automatic
HasCoequalizer.coequalizer_π_stalk_isLocalRingHom _ _⟩
(LocallyRingedSpace.Hom.ext _ _ (coequalizer.condition f.1 g.1))
#align algebraic_geometry.LocallyRingedSpace.coequalizer_cofork AlgebraicGeometry.LocallyRingedSpace.coequalizerCofork
theorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
(h : IsLocalRingHom (PresheafedSpace.stalkMap f x)) :
IsLocalRingHom (PresheafedSpace.stalkMap g x) := by
rw [PresheafedSpace.stalkMap.congr_hom _ _ H.symm x]; infer_instance
#align algebraic_geometry.LocallyRingedSpace.is_local_ring_hom_stalk_map_congr AlgebraicGeometry.LocallyRingedSpace.isLocalRingHom_stalkMap_congr
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
set h := _
change IsLocalRingHom h
suffices : IsLocalRingHom ((PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _).comp h)
· apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _)
change IsLocalRingHom (_ ≫ PresheafedSpace.stalkMap (coequalizerCofork f g).π.val y)
erw [← PresheafedSpace.stalkMap.comp]
apply isLocalRingHom_stalkMap_congr _ _ (coequalizer.π_desc s.π.1 e).symm y
infer_instance
|
constructor
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
set h := _
change IsLocalRingHom h
suffices : IsLocalRingHom ((PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _).comp h)
· apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _)
change IsLocalRingHom (_ ≫ PresheafedSpace.stalkMap (coequalizerCofork f g).π.val y)
erw [← PresheafedSpace.stalkMap.comp]
apply isLocalRingHom_stalkMap_congr _ _ (coequalizer.π_desc s.π.1 e).symm y
infer_instance
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case create.refine_2.left
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
⊢ Cofork.π (coequalizerCofork f g) ≫
{ val := coequalizer.desc (Cofork.π s).val e,
prop :=
(_ :
∀ (x : ↑↑(coequalizerCofork f g).pt.toPresheafedSpace),
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.desc (Cofork.π s).val e) x)) } =
Cofork.π s
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]
apply RingHom.isUnit_map
rw [← isUnit_map_iff ((coequalizer.π f.val g.val : _).c.app _), ← comp_apply,
NatTrans.naturality, comp_apply, TopCat.Presheaf.pushforwardObj_map, ←
isUnit_map_iff (Y.presheaf.map (eqToHom hV').op)]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← comp_apply, ← Y.presheaf.map_comp]
convert @RingedSpace.isUnit_res_basicOpen Y.toRingedSpace (unop _)
(((coequalizer.π f.val g.val).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_stalk_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_stalk_isLocalRingHom
end HasCoequalizer
/-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. -/
noncomputable def coequalizer : LocallyRingedSpace where
toSheafedSpace := Limits.coequalizer f.1 g.1
localRing x := by
obtain ⟨y, rfl⟩ :=
(TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x
exact (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) y).domain_localRing
#align algebraic_geometry.LocallyRingedSpace.coequalizer AlgebraicGeometry.LocallyRingedSpace.coequalizer
/-- The explicit coequalizer cofork of locally ringed spaces. -/
noncomputable def coequalizerCofork : Cofork f g :=
@Cofork.ofπ _ _ _ _ f g (coequalizer f g) ⟨coequalizer.π f.1 g.1,
-- Porting note : this used to be automatic
HasCoequalizer.coequalizer_π_stalk_isLocalRingHom _ _⟩
(LocallyRingedSpace.Hom.ext _ _ (coequalizer.condition f.1 g.1))
#align algebraic_geometry.LocallyRingedSpace.coequalizer_cofork AlgebraicGeometry.LocallyRingedSpace.coequalizerCofork
theorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
(h : IsLocalRingHom (PresheafedSpace.stalkMap f x)) :
IsLocalRingHom (PresheafedSpace.stalkMap g x) := by
rw [PresheafedSpace.stalkMap.congr_hom _ _ H.symm x]; infer_instance
#align algebraic_geometry.LocallyRingedSpace.is_local_ring_hom_stalk_map_congr AlgebraicGeometry.LocallyRingedSpace.isLocalRingHom_stalkMap_congr
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
set h := _
change IsLocalRingHom h
suffices : IsLocalRingHom ((PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _).comp h)
· apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _)
change IsLocalRingHom (_ ≫ PresheafedSpace.stalkMap (coequalizerCofork f g).π.val y)
erw [← PresheafedSpace.stalkMap.comp]
apply isLocalRingHom_stalkMap_congr _ _ (coequalizer.π_desc s.π.1 e).symm y
infer_instance
constructor
·
|
exact LocallyRingedSpace.Hom.ext _ _ (coequalizer.π_desc _ _)
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
set h := _
change IsLocalRingHom h
suffices : IsLocalRingHom ((PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _).comp h)
· apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _)
change IsLocalRingHom (_ ≫ PresheafedSpace.stalkMap (coequalizerCofork f g).π.val y)
erw [← PresheafedSpace.stalkMap.comp]
apply isLocalRingHom_stalkMap_congr _ _ (coequalizer.π_desc s.π.1 e).symm y
infer_instance
constructor
·
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case create.refine_2.right
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
⊢ ∀
{m :
((Functor.const WalkingParallelPair).obj (coequalizerCofork f g).pt).obj WalkingParallelPair.one ⟶
((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.one},
Cofork.π (coequalizerCofork f g) ≫ m = Cofork.π s →
m =
{ val := coequalizer.desc (Cofork.π s).val e,
prop :=
(_ :
∀ (x : ↑↑(coequalizerCofork f g).pt.toPresheafedSpace),
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.desc (Cofork.π s).val e) x)) }
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]
apply RingHom.isUnit_map
rw [← isUnit_map_iff ((coequalizer.π f.val g.val : _).c.app _), ← comp_apply,
NatTrans.naturality, comp_apply, TopCat.Presheaf.pushforwardObj_map, ←
isUnit_map_iff (Y.presheaf.map (eqToHom hV').op)]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← comp_apply, ← Y.presheaf.map_comp]
convert @RingedSpace.isUnit_res_basicOpen Y.toRingedSpace (unop _)
(((coequalizer.π f.val g.val).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_stalk_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_stalk_isLocalRingHom
end HasCoequalizer
/-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. -/
noncomputable def coequalizer : LocallyRingedSpace where
toSheafedSpace := Limits.coequalizer f.1 g.1
localRing x := by
obtain ⟨y, rfl⟩ :=
(TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x
exact (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) y).domain_localRing
#align algebraic_geometry.LocallyRingedSpace.coequalizer AlgebraicGeometry.LocallyRingedSpace.coequalizer
/-- The explicit coequalizer cofork of locally ringed spaces. -/
noncomputable def coequalizerCofork : Cofork f g :=
@Cofork.ofπ _ _ _ _ f g (coequalizer f g) ⟨coequalizer.π f.1 g.1,
-- Porting note : this used to be automatic
HasCoequalizer.coequalizer_π_stalk_isLocalRingHom _ _⟩
(LocallyRingedSpace.Hom.ext _ _ (coequalizer.condition f.1 g.1))
#align algebraic_geometry.LocallyRingedSpace.coequalizer_cofork AlgebraicGeometry.LocallyRingedSpace.coequalizerCofork
theorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
(h : IsLocalRingHom (PresheafedSpace.stalkMap f x)) :
IsLocalRingHom (PresheafedSpace.stalkMap g x) := by
rw [PresheafedSpace.stalkMap.congr_hom _ _ H.symm x]; infer_instance
#align algebraic_geometry.LocallyRingedSpace.is_local_ring_hom_stalk_map_congr AlgebraicGeometry.LocallyRingedSpace.isLocalRingHom_stalkMap_congr
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
set h := _
change IsLocalRingHom h
suffices : IsLocalRingHom ((PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _).comp h)
· apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _)
change IsLocalRingHom (_ ≫ PresheafedSpace.stalkMap (coequalizerCofork f g).π.val y)
erw [← PresheafedSpace.stalkMap.comp]
apply isLocalRingHom_stalkMap_congr _ _ (coequalizer.π_desc s.π.1 e).symm y
infer_instance
constructor
· exact LocallyRingedSpace.Hom.ext _ _ (coequalizer.π_desc _ _)
|
intro m h
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
set h := _
change IsLocalRingHom h
suffices : IsLocalRingHom ((PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _).comp h)
· apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _)
change IsLocalRingHom (_ ≫ PresheafedSpace.stalkMap (coequalizerCofork f g).π.val y)
erw [← PresheafedSpace.stalkMap.comp]
apply isLocalRingHom_stalkMap_congr _ _ (coequalizer.π_desc s.π.1 e).symm y
infer_instance
constructor
· exact LocallyRingedSpace.Hom.ext _ _ (coequalizer.π_desc _ _)
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case create.refine_2.right
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
m :
((Functor.const WalkingParallelPair).obj (coequalizerCofork f g).pt).obj WalkingParallelPair.one ⟶
((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.one
h : Cofork.π (coequalizerCofork f g) ≫ m = Cofork.π s
⊢ m =
{ val := coequalizer.desc (Cofork.π s).val e,
prop :=
(_ :
∀ (x : ↑↑(coequalizerCofork f g).pt.toPresheafedSpace),
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.desc (Cofork.π s).val e) x)) }
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]
apply RingHom.isUnit_map
rw [← isUnit_map_iff ((coequalizer.π f.val g.val : _).c.app _), ← comp_apply,
NatTrans.naturality, comp_apply, TopCat.Presheaf.pushforwardObj_map, ←
isUnit_map_iff (Y.presheaf.map (eqToHom hV').op)]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← comp_apply, ← Y.presheaf.map_comp]
convert @RingedSpace.isUnit_res_basicOpen Y.toRingedSpace (unop _)
(((coequalizer.π f.val g.val).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_stalk_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_stalk_isLocalRingHom
end HasCoequalizer
/-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. -/
noncomputable def coequalizer : LocallyRingedSpace where
toSheafedSpace := Limits.coequalizer f.1 g.1
localRing x := by
obtain ⟨y, rfl⟩ :=
(TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x
exact (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) y).domain_localRing
#align algebraic_geometry.LocallyRingedSpace.coequalizer AlgebraicGeometry.LocallyRingedSpace.coequalizer
/-- The explicit coequalizer cofork of locally ringed spaces. -/
noncomputable def coequalizerCofork : Cofork f g :=
@Cofork.ofπ _ _ _ _ f g (coequalizer f g) ⟨coequalizer.π f.1 g.1,
-- Porting note : this used to be automatic
HasCoequalizer.coequalizer_π_stalk_isLocalRingHom _ _⟩
(LocallyRingedSpace.Hom.ext _ _ (coequalizer.condition f.1 g.1))
#align algebraic_geometry.LocallyRingedSpace.coequalizer_cofork AlgebraicGeometry.LocallyRingedSpace.coequalizerCofork
theorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
(h : IsLocalRingHom (PresheafedSpace.stalkMap f x)) :
IsLocalRingHom (PresheafedSpace.stalkMap g x) := by
rw [PresheafedSpace.stalkMap.congr_hom _ _ H.symm x]; infer_instance
#align algebraic_geometry.LocallyRingedSpace.is_local_ring_hom_stalk_map_congr AlgebraicGeometry.LocallyRingedSpace.isLocalRingHom_stalkMap_congr
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
set h := _
change IsLocalRingHom h
suffices : IsLocalRingHom ((PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _).comp h)
· apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _)
change IsLocalRingHom (_ ≫ PresheafedSpace.stalkMap (coequalizerCofork f g).π.val y)
erw [← PresheafedSpace.stalkMap.comp]
apply isLocalRingHom_stalkMap_congr _ _ (coequalizer.π_desc s.π.1 e).symm y
infer_instance
constructor
· exact LocallyRingedSpace.Hom.ext _ _ (coequalizer.π_desc _ _)
intro m h
|
replace h : (coequalizerCofork f g).π.1 ≫ m.1 = s.π.1 := by rw [← h]; rfl
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
set h := _
change IsLocalRingHom h
suffices : IsLocalRingHom ((PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _).comp h)
· apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _)
change IsLocalRingHom (_ ≫ PresheafedSpace.stalkMap (coequalizerCofork f g).π.val y)
erw [← PresheafedSpace.stalkMap.comp]
apply isLocalRingHom_stalkMap_congr _ _ (coequalizer.π_desc s.π.1 e).symm y
infer_instance
constructor
· exact LocallyRingedSpace.Hom.ext _ _ (coequalizer.π_desc _ _)
intro m h
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
m :
((Functor.const WalkingParallelPair).obj (coequalizerCofork f g).pt).obj WalkingParallelPair.one ⟶
((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.one
h : Cofork.π (coequalizerCofork f g) ≫ m = Cofork.π s
⊢ (Cofork.π (coequalizerCofork f g)).val ≫ m.val = (Cofork.π s).val
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]
apply RingHom.isUnit_map
rw [← isUnit_map_iff ((coequalizer.π f.val g.val : _).c.app _), ← comp_apply,
NatTrans.naturality, comp_apply, TopCat.Presheaf.pushforwardObj_map, ←
isUnit_map_iff (Y.presheaf.map (eqToHom hV').op)]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← comp_apply, ← Y.presheaf.map_comp]
convert @RingedSpace.isUnit_res_basicOpen Y.toRingedSpace (unop _)
(((coequalizer.π f.val g.val).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_stalk_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_stalk_isLocalRingHom
end HasCoequalizer
/-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. -/
noncomputable def coequalizer : LocallyRingedSpace where
toSheafedSpace := Limits.coequalizer f.1 g.1
localRing x := by
obtain ⟨y, rfl⟩ :=
(TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x
exact (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) y).domain_localRing
#align algebraic_geometry.LocallyRingedSpace.coequalizer AlgebraicGeometry.LocallyRingedSpace.coequalizer
/-- The explicit coequalizer cofork of locally ringed spaces. -/
noncomputable def coequalizerCofork : Cofork f g :=
@Cofork.ofπ _ _ _ _ f g (coequalizer f g) ⟨coequalizer.π f.1 g.1,
-- Porting note : this used to be automatic
HasCoequalizer.coequalizer_π_stalk_isLocalRingHom _ _⟩
(LocallyRingedSpace.Hom.ext _ _ (coequalizer.condition f.1 g.1))
#align algebraic_geometry.LocallyRingedSpace.coequalizer_cofork AlgebraicGeometry.LocallyRingedSpace.coequalizerCofork
theorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
(h : IsLocalRingHom (PresheafedSpace.stalkMap f x)) :
IsLocalRingHom (PresheafedSpace.stalkMap g x) := by
rw [PresheafedSpace.stalkMap.congr_hom _ _ H.symm x]; infer_instance
#align algebraic_geometry.LocallyRingedSpace.is_local_ring_hom_stalk_map_congr AlgebraicGeometry.LocallyRingedSpace.isLocalRingHom_stalkMap_congr
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
set h := _
change IsLocalRingHom h
suffices : IsLocalRingHom ((PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _).comp h)
· apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _)
change IsLocalRingHom (_ ≫ PresheafedSpace.stalkMap (coequalizerCofork f g).π.val y)
erw [← PresheafedSpace.stalkMap.comp]
apply isLocalRingHom_stalkMap_congr _ _ (coequalizer.π_desc s.π.1 e).symm y
infer_instance
constructor
· exact LocallyRingedSpace.Hom.ext _ _ (coequalizer.π_desc _ _)
intro m h
replace h : (coequalizerCofork f g).π.1 ≫ m.1 = s.π.1 := by
|
rw [← h]
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
set h := _
change IsLocalRingHom h
suffices : IsLocalRingHom ((PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _).comp h)
· apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _)
change IsLocalRingHom (_ ≫ PresheafedSpace.stalkMap (coequalizerCofork f g).π.val y)
erw [← PresheafedSpace.stalkMap.comp]
apply isLocalRingHom_stalkMap_congr _ _ (coequalizer.π_desc s.π.1 e).symm y
infer_instance
constructor
· exact LocallyRingedSpace.Hom.ext _ _ (coequalizer.π_desc _ _)
intro m h
replace h : (coequalizerCofork f g).π.1 ≫ m.1 = s.π.1 := by
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
m :
((Functor.const WalkingParallelPair).obj (coequalizerCofork f g).pt).obj WalkingParallelPair.one ⟶
((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.one
h : Cofork.π (coequalizerCofork f g) ≫ m = Cofork.π s
⊢ (Cofork.π (coequalizerCofork f g)).val ≫ m.val = (Cofork.π (coequalizerCofork f g) ≫ m).val
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]
apply RingHom.isUnit_map
rw [← isUnit_map_iff ((coequalizer.π f.val g.val : _).c.app _), ← comp_apply,
NatTrans.naturality, comp_apply, TopCat.Presheaf.pushforwardObj_map, ←
isUnit_map_iff (Y.presheaf.map (eqToHom hV').op)]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← comp_apply, ← Y.presheaf.map_comp]
convert @RingedSpace.isUnit_res_basicOpen Y.toRingedSpace (unop _)
(((coequalizer.π f.val g.val).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_stalk_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_stalk_isLocalRingHom
end HasCoequalizer
/-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. -/
noncomputable def coequalizer : LocallyRingedSpace where
toSheafedSpace := Limits.coequalizer f.1 g.1
localRing x := by
obtain ⟨y, rfl⟩ :=
(TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x
exact (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) y).domain_localRing
#align algebraic_geometry.LocallyRingedSpace.coequalizer AlgebraicGeometry.LocallyRingedSpace.coequalizer
/-- The explicit coequalizer cofork of locally ringed spaces. -/
noncomputable def coequalizerCofork : Cofork f g :=
@Cofork.ofπ _ _ _ _ f g (coequalizer f g) ⟨coequalizer.π f.1 g.1,
-- Porting note : this used to be automatic
HasCoequalizer.coequalizer_π_stalk_isLocalRingHom _ _⟩
(LocallyRingedSpace.Hom.ext _ _ (coequalizer.condition f.1 g.1))
#align algebraic_geometry.LocallyRingedSpace.coequalizer_cofork AlgebraicGeometry.LocallyRingedSpace.coequalizerCofork
theorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
(h : IsLocalRingHom (PresheafedSpace.stalkMap f x)) :
IsLocalRingHom (PresheafedSpace.stalkMap g x) := by
rw [PresheafedSpace.stalkMap.congr_hom _ _ H.symm x]; infer_instance
#align algebraic_geometry.LocallyRingedSpace.is_local_ring_hom_stalk_map_congr AlgebraicGeometry.LocallyRingedSpace.isLocalRingHom_stalkMap_congr
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
set h := _
change IsLocalRingHom h
suffices : IsLocalRingHom ((PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _).comp h)
· apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _)
change IsLocalRingHom (_ ≫ PresheafedSpace.stalkMap (coequalizerCofork f g).π.val y)
erw [← PresheafedSpace.stalkMap.comp]
apply isLocalRingHom_stalkMap_congr _ _ (coequalizer.π_desc s.π.1 e).symm y
infer_instance
constructor
· exact LocallyRingedSpace.Hom.ext _ _ (coequalizer.π_desc _ _)
intro m h
replace h : (coequalizerCofork f g).π.1 ≫ m.1 = s.π.1 := by rw [← h];
|
rfl
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
set h := _
change IsLocalRingHom h
suffices : IsLocalRingHom ((PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _).comp h)
· apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _)
change IsLocalRingHom (_ ≫ PresheafedSpace.stalkMap (coequalizerCofork f g).π.val y)
erw [← PresheafedSpace.stalkMap.comp]
apply isLocalRingHom_stalkMap_congr _ _ (coequalizer.π_desc s.π.1 e).symm y
infer_instance
constructor
· exact LocallyRingedSpace.Hom.ext _ _ (coequalizer.π_desc _ _)
intro m h
replace h : (coequalizerCofork f g).π.1 ≫ m.1 = s.π.1 := by rw [← h];
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case create.refine_2.right
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
m :
((Functor.const WalkingParallelPair).obj (coequalizerCofork f g).pt).obj WalkingParallelPair.one ⟶
((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.one
h : (Cofork.π (coequalizerCofork f g)).val ≫ m.val = (Cofork.π s).val
⊢ m =
{ val := coequalizer.desc (Cofork.π s).val e,
prop :=
(_ :
∀ (x : ↑↑(coequalizerCofork f g).pt.toPresheafedSpace),
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.desc (Cofork.π s).val e) x)) }
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]
apply RingHom.isUnit_map
rw [← isUnit_map_iff ((coequalizer.π f.val g.val : _).c.app _), ← comp_apply,
NatTrans.naturality, comp_apply, TopCat.Presheaf.pushforwardObj_map, ←
isUnit_map_iff (Y.presheaf.map (eqToHom hV').op)]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← comp_apply, ← Y.presheaf.map_comp]
convert @RingedSpace.isUnit_res_basicOpen Y.toRingedSpace (unop _)
(((coequalizer.π f.val g.val).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_stalk_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_stalk_isLocalRingHom
end HasCoequalizer
/-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. -/
noncomputable def coequalizer : LocallyRingedSpace where
toSheafedSpace := Limits.coequalizer f.1 g.1
localRing x := by
obtain ⟨y, rfl⟩ :=
(TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x
exact (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) y).domain_localRing
#align algebraic_geometry.LocallyRingedSpace.coequalizer AlgebraicGeometry.LocallyRingedSpace.coequalizer
/-- The explicit coequalizer cofork of locally ringed spaces. -/
noncomputable def coequalizerCofork : Cofork f g :=
@Cofork.ofπ _ _ _ _ f g (coequalizer f g) ⟨coequalizer.π f.1 g.1,
-- Porting note : this used to be automatic
HasCoequalizer.coequalizer_π_stalk_isLocalRingHom _ _⟩
(LocallyRingedSpace.Hom.ext _ _ (coequalizer.condition f.1 g.1))
#align algebraic_geometry.LocallyRingedSpace.coequalizer_cofork AlgebraicGeometry.LocallyRingedSpace.coequalizerCofork
theorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
(h : IsLocalRingHom (PresheafedSpace.stalkMap f x)) :
IsLocalRingHom (PresheafedSpace.stalkMap g x) := by
rw [PresheafedSpace.stalkMap.congr_hom _ _ H.symm x]; infer_instance
#align algebraic_geometry.LocallyRingedSpace.is_local_ring_hom_stalk_map_congr AlgebraicGeometry.LocallyRingedSpace.isLocalRingHom_stalkMap_congr
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
set h := _
change IsLocalRingHom h
suffices : IsLocalRingHom ((PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _).comp h)
· apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _)
change IsLocalRingHom (_ ≫ PresheafedSpace.stalkMap (coequalizerCofork f g).π.val y)
erw [← PresheafedSpace.stalkMap.comp]
apply isLocalRingHom_stalkMap_congr _ _ (coequalizer.π_desc s.π.1 e).symm y
infer_instance
constructor
· exact LocallyRingedSpace.Hom.ext _ _ (coequalizer.π_desc _ _)
intro m h
replace h : (coequalizerCofork f g).π.1 ≫ m.1 = s.π.1 := by rw [← h]; rfl
|
apply LocallyRingedSpace.Hom.ext
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
set h := _
change IsLocalRingHom h
suffices : IsLocalRingHom ((PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _).comp h)
· apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _)
change IsLocalRingHom (_ ≫ PresheafedSpace.stalkMap (coequalizerCofork f g).π.val y)
erw [← PresheafedSpace.stalkMap.comp]
apply isLocalRingHom_stalkMap_congr _ _ (coequalizer.π_desc s.π.1 e).symm y
infer_instance
constructor
· exact LocallyRingedSpace.Hom.ext _ _ (coequalizer.π_desc _ _)
intro m h
replace h : (coequalizerCofork f g).π.1 ≫ m.1 = s.π.1 := by rw [← h]; rfl
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case create.refine_2.right.val
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
m :
((Functor.const WalkingParallelPair).obj (coequalizerCofork f g).pt).obj WalkingParallelPair.one ⟶
((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.one
h : (Cofork.π (coequalizerCofork f g)).val ≫ m.val = (Cofork.π s).val
⊢ m.val =
{ val := coequalizer.desc (Cofork.π s).val e,
prop :=
(_ :
∀ (x : ↑↑(coequalizerCofork f g).pt.toPresheafedSpace),
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.desc (Cofork.π s).val e) x)) }.val
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]
apply RingHom.isUnit_map
rw [← isUnit_map_iff ((coequalizer.π f.val g.val : _).c.app _), ← comp_apply,
NatTrans.naturality, comp_apply, TopCat.Presheaf.pushforwardObj_map, ←
isUnit_map_iff (Y.presheaf.map (eqToHom hV').op)]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← comp_apply, ← Y.presheaf.map_comp]
convert @RingedSpace.isUnit_res_basicOpen Y.toRingedSpace (unop _)
(((coequalizer.π f.val g.val).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_stalk_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_stalk_isLocalRingHom
end HasCoequalizer
/-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. -/
noncomputable def coequalizer : LocallyRingedSpace where
toSheafedSpace := Limits.coequalizer f.1 g.1
localRing x := by
obtain ⟨y, rfl⟩ :=
(TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x
exact (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) y).domain_localRing
#align algebraic_geometry.LocallyRingedSpace.coequalizer AlgebraicGeometry.LocallyRingedSpace.coequalizer
/-- The explicit coequalizer cofork of locally ringed spaces. -/
noncomputable def coequalizerCofork : Cofork f g :=
@Cofork.ofπ _ _ _ _ f g (coequalizer f g) ⟨coequalizer.π f.1 g.1,
-- Porting note : this used to be automatic
HasCoequalizer.coequalizer_π_stalk_isLocalRingHom _ _⟩
(LocallyRingedSpace.Hom.ext _ _ (coequalizer.condition f.1 g.1))
#align algebraic_geometry.LocallyRingedSpace.coequalizer_cofork AlgebraicGeometry.LocallyRingedSpace.coequalizerCofork
theorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
(h : IsLocalRingHom (PresheafedSpace.stalkMap f x)) :
IsLocalRingHom (PresheafedSpace.stalkMap g x) := by
rw [PresheafedSpace.stalkMap.congr_hom _ _ H.symm x]; infer_instance
#align algebraic_geometry.LocallyRingedSpace.is_local_ring_hom_stalk_map_congr AlgebraicGeometry.LocallyRingedSpace.isLocalRingHom_stalkMap_congr
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
set h := _
change IsLocalRingHom h
suffices : IsLocalRingHom ((PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _).comp h)
· apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _)
change IsLocalRingHom (_ ≫ PresheafedSpace.stalkMap (coequalizerCofork f g).π.val y)
erw [← PresheafedSpace.stalkMap.comp]
apply isLocalRingHom_stalkMap_congr _ _ (coequalizer.π_desc s.π.1 e).symm y
infer_instance
constructor
· exact LocallyRingedSpace.Hom.ext _ _ (coequalizer.π_desc _ _)
intro m h
replace h : (coequalizerCofork f g).π.1 ≫ m.1 = s.π.1 := by rw [← h]; rfl
apply LocallyRingedSpace.Hom.ext
|
apply (colimit.isColimit (parallelPair f.1 g.1)).uniq (Cofork.ofπ s.π.1 e) m.1
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
set h := _
change IsLocalRingHom h
suffices : IsLocalRingHom ((PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _).comp h)
· apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _)
change IsLocalRingHom (_ ≫ PresheafedSpace.stalkMap (coequalizerCofork f g).π.val y)
erw [← PresheafedSpace.stalkMap.comp]
apply isLocalRingHom_stalkMap_congr _ _ (coequalizer.π_desc s.π.1 e).symm y
infer_instance
constructor
· exact LocallyRingedSpace.Hom.ext _ _ (coequalizer.π_desc _ _)
intro m h
replace h : (coequalizerCofork f g).π.1 ≫ m.1 = s.π.1 := by rw [← h]; rfl
apply LocallyRingedSpace.Hom.ext
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case create.refine_2.right.val
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
m :
((Functor.const WalkingParallelPair).obj (coequalizerCofork f g).pt).obj WalkingParallelPair.one ⟶
((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.one
h : (Cofork.π (coequalizerCofork f g)).val ≫ m.val = (Cofork.π s).val
⊢ ∀ (j : WalkingParallelPair),
(colimit.cocone (parallelPair f.val g.val)).ι.app j ≫ m.val = (Cofork.ofπ (Cofork.π s).val e).ι.app j
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]
apply RingHom.isUnit_map
rw [← isUnit_map_iff ((coequalizer.π f.val g.val : _).c.app _), ← comp_apply,
NatTrans.naturality, comp_apply, TopCat.Presheaf.pushforwardObj_map, ←
isUnit_map_iff (Y.presheaf.map (eqToHom hV').op)]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← comp_apply, ← Y.presheaf.map_comp]
convert @RingedSpace.isUnit_res_basicOpen Y.toRingedSpace (unop _)
(((coequalizer.π f.val g.val).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_stalk_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_stalk_isLocalRingHom
end HasCoequalizer
/-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. -/
noncomputable def coequalizer : LocallyRingedSpace where
toSheafedSpace := Limits.coequalizer f.1 g.1
localRing x := by
obtain ⟨y, rfl⟩ :=
(TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x
exact (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) y).domain_localRing
#align algebraic_geometry.LocallyRingedSpace.coequalizer AlgebraicGeometry.LocallyRingedSpace.coequalizer
/-- The explicit coequalizer cofork of locally ringed spaces. -/
noncomputable def coequalizerCofork : Cofork f g :=
@Cofork.ofπ _ _ _ _ f g (coequalizer f g) ⟨coequalizer.π f.1 g.1,
-- Porting note : this used to be automatic
HasCoequalizer.coequalizer_π_stalk_isLocalRingHom _ _⟩
(LocallyRingedSpace.Hom.ext _ _ (coequalizer.condition f.1 g.1))
#align algebraic_geometry.LocallyRingedSpace.coequalizer_cofork AlgebraicGeometry.LocallyRingedSpace.coequalizerCofork
theorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
(h : IsLocalRingHom (PresheafedSpace.stalkMap f x)) :
IsLocalRingHom (PresheafedSpace.stalkMap g x) := by
rw [PresheafedSpace.stalkMap.congr_hom _ _ H.symm x]; infer_instance
#align algebraic_geometry.LocallyRingedSpace.is_local_ring_hom_stalk_map_congr AlgebraicGeometry.LocallyRingedSpace.isLocalRingHom_stalkMap_congr
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
set h := _
change IsLocalRingHom h
suffices : IsLocalRingHom ((PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _).comp h)
· apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _)
change IsLocalRingHom (_ ≫ PresheafedSpace.stalkMap (coequalizerCofork f g).π.val y)
erw [← PresheafedSpace.stalkMap.comp]
apply isLocalRingHom_stalkMap_congr _ _ (coequalizer.π_desc s.π.1 e).symm y
infer_instance
constructor
· exact LocallyRingedSpace.Hom.ext _ _ (coequalizer.π_desc _ _)
intro m h
replace h : (coequalizerCofork f g).π.1 ≫ m.1 = s.π.1 := by rw [← h]; rfl
apply LocallyRingedSpace.Hom.ext
apply (colimit.isColimit (parallelPair f.1 g.1)).uniq (Cofork.ofπ s.π.1 e) m.1
|
rintro ⟨⟩
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
set h := _
change IsLocalRingHom h
suffices : IsLocalRingHom ((PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _).comp h)
· apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _)
change IsLocalRingHom (_ ≫ PresheafedSpace.stalkMap (coequalizerCofork f g).π.val y)
erw [← PresheafedSpace.stalkMap.comp]
apply isLocalRingHom_stalkMap_congr _ _ (coequalizer.π_desc s.π.1 e).symm y
infer_instance
constructor
· exact LocallyRingedSpace.Hom.ext _ _ (coequalizer.π_desc _ _)
intro m h
replace h : (coequalizerCofork f g).π.1 ≫ m.1 = s.π.1 := by rw [← h]; rfl
apply LocallyRingedSpace.Hom.ext
apply (colimit.isColimit (parallelPair f.1 g.1)).uniq (Cofork.ofπ s.π.1 e) m.1
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case create.refine_2.right.val.zero
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
m :
((Functor.const WalkingParallelPair).obj (coequalizerCofork f g).pt).obj WalkingParallelPair.one ⟶
((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.one
h : (Cofork.π (coequalizerCofork f g)).val ≫ m.val = (Cofork.π s).val
⊢ (colimit.cocone (parallelPair f.val g.val)).ι.app WalkingParallelPair.zero ≫ m.val =
(Cofork.ofπ (Cofork.π s).val e).ι.app WalkingParallelPair.zero
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]
apply RingHom.isUnit_map
rw [← isUnit_map_iff ((coequalizer.π f.val g.val : _).c.app _), ← comp_apply,
NatTrans.naturality, comp_apply, TopCat.Presheaf.pushforwardObj_map, ←
isUnit_map_iff (Y.presheaf.map (eqToHom hV').op)]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← comp_apply, ← Y.presheaf.map_comp]
convert @RingedSpace.isUnit_res_basicOpen Y.toRingedSpace (unop _)
(((coequalizer.π f.val g.val).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_stalk_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_stalk_isLocalRingHom
end HasCoequalizer
/-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. -/
noncomputable def coequalizer : LocallyRingedSpace where
toSheafedSpace := Limits.coequalizer f.1 g.1
localRing x := by
obtain ⟨y, rfl⟩ :=
(TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x
exact (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) y).domain_localRing
#align algebraic_geometry.LocallyRingedSpace.coequalizer AlgebraicGeometry.LocallyRingedSpace.coequalizer
/-- The explicit coequalizer cofork of locally ringed spaces. -/
noncomputable def coequalizerCofork : Cofork f g :=
@Cofork.ofπ _ _ _ _ f g (coequalizer f g) ⟨coequalizer.π f.1 g.1,
-- Porting note : this used to be automatic
HasCoequalizer.coequalizer_π_stalk_isLocalRingHom _ _⟩
(LocallyRingedSpace.Hom.ext _ _ (coequalizer.condition f.1 g.1))
#align algebraic_geometry.LocallyRingedSpace.coequalizer_cofork AlgebraicGeometry.LocallyRingedSpace.coequalizerCofork
theorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
(h : IsLocalRingHom (PresheafedSpace.stalkMap f x)) :
IsLocalRingHom (PresheafedSpace.stalkMap g x) := by
rw [PresheafedSpace.stalkMap.congr_hom _ _ H.symm x]; infer_instance
#align algebraic_geometry.LocallyRingedSpace.is_local_ring_hom_stalk_map_congr AlgebraicGeometry.LocallyRingedSpace.isLocalRingHom_stalkMap_congr
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
set h := _
change IsLocalRingHom h
suffices : IsLocalRingHom ((PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _).comp h)
· apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _)
change IsLocalRingHom (_ ≫ PresheafedSpace.stalkMap (coequalizerCofork f g).π.val y)
erw [← PresheafedSpace.stalkMap.comp]
apply isLocalRingHom_stalkMap_congr _ _ (coequalizer.π_desc s.π.1 e).symm y
infer_instance
constructor
· exact LocallyRingedSpace.Hom.ext _ _ (coequalizer.π_desc _ _)
intro m h
replace h : (coequalizerCofork f g).π.1 ≫ m.1 = s.π.1 := by rw [← h]; rfl
apply LocallyRingedSpace.Hom.ext
apply (colimit.isColimit (parallelPair f.1 g.1)).uniq (Cofork.ofπ s.π.1 e) m.1
rintro ⟨⟩
·
|
rw [← (colimit.cocone (parallelPair f.val g.val)).w WalkingParallelPairHom.left,
Category.assoc]
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
set h := _
change IsLocalRingHom h
suffices : IsLocalRingHom ((PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _).comp h)
· apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _)
change IsLocalRingHom (_ ≫ PresheafedSpace.stalkMap (coequalizerCofork f g).π.val y)
erw [← PresheafedSpace.stalkMap.comp]
apply isLocalRingHom_stalkMap_congr _ _ (coequalizer.π_desc s.π.1 e).symm y
infer_instance
constructor
· exact LocallyRingedSpace.Hom.ext _ _ (coequalizer.π_desc _ _)
intro m h
replace h : (coequalizerCofork f g).π.1 ≫ m.1 = s.π.1 := by rw [← h]; rfl
apply LocallyRingedSpace.Hom.ext
apply (colimit.isColimit (parallelPair f.1 g.1)).uniq (Cofork.ofπ s.π.1 e) m.1
rintro ⟨⟩
·
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case create.refine_2.right.val.zero
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
m :
((Functor.const WalkingParallelPair).obj (coequalizerCofork f g).pt).obj WalkingParallelPair.one ⟶
((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.one
h : (Cofork.π (coequalizerCofork f g)).val ≫ m.val = (Cofork.π s).val
⊢ (parallelPair f.val g.val).map WalkingParallelPairHom.left ≫
(colimit.cocone (parallelPair f.val g.val)).ι.app WalkingParallelPair.one ≫ m.val =
(Cofork.ofπ (Cofork.π s).val e).ι.app WalkingParallelPair.zero
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]
apply RingHom.isUnit_map
rw [← isUnit_map_iff ((coequalizer.π f.val g.val : _).c.app _), ← comp_apply,
NatTrans.naturality, comp_apply, TopCat.Presheaf.pushforwardObj_map, ←
isUnit_map_iff (Y.presheaf.map (eqToHom hV').op)]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← comp_apply, ← Y.presheaf.map_comp]
convert @RingedSpace.isUnit_res_basicOpen Y.toRingedSpace (unop _)
(((coequalizer.π f.val g.val).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_stalk_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_stalk_isLocalRingHom
end HasCoequalizer
/-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. -/
noncomputable def coequalizer : LocallyRingedSpace where
toSheafedSpace := Limits.coequalizer f.1 g.1
localRing x := by
obtain ⟨y, rfl⟩ :=
(TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x
exact (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) y).domain_localRing
#align algebraic_geometry.LocallyRingedSpace.coequalizer AlgebraicGeometry.LocallyRingedSpace.coequalizer
/-- The explicit coequalizer cofork of locally ringed spaces. -/
noncomputable def coequalizerCofork : Cofork f g :=
@Cofork.ofπ _ _ _ _ f g (coequalizer f g) ⟨coequalizer.π f.1 g.1,
-- Porting note : this used to be automatic
HasCoequalizer.coequalizer_π_stalk_isLocalRingHom _ _⟩
(LocallyRingedSpace.Hom.ext _ _ (coequalizer.condition f.1 g.1))
#align algebraic_geometry.LocallyRingedSpace.coequalizer_cofork AlgebraicGeometry.LocallyRingedSpace.coequalizerCofork
theorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
(h : IsLocalRingHom (PresheafedSpace.stalkMap f x)) :
IsLocalRingHom (PresheafedSpace.stalkMap g x) := by
rw [PresheafedSpace.stalkMap.congr_hom _ _ H.symm x]; infer_instance
#align algebraic_geometry.LocallyRingedSpace.is_local_ring_hom_stalk_map_congr AlgebraicGeometry.LocallyRingedSpace.isLocalRingHom_stalkMap_congr
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
set h := _
change IsLocalRingHom h
suffices : IsLocalRingHom ((PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _).comp h)
· apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _)
change IsLocalRingHom (_ ≫ PresheafedSpace.stalkMap (coequalizerCofork f g).π.val y)
erw [← PresheafedSpace.stalkMap.comp]
apply isLocalRingHom_stalkMap_congr _ _ (coequalizer.π_desc s.π.1 e).symm y
infer_instance
constructor
· exact LocallyRingedSpace.Hom.ext _ _ (coequalizer.π_desc _ _)
intro m h
replace h : (coequalizerCofork f g).π.1 ≫ m.1 = s.π.1 := by rw [← h]; rfl
apply LocallyRingedSpace.Hom.ext
apply (colimit.isColimit (parallelPair f.1 g.1)).uniq (Cofork.ofπ s.π.1 e) m.1
rintro ⟨⟩
· rw [← (colimit.cocone (parallelPair f.val g.val)).w WalkingParallelPairHom.left,
Category.assoc]
|
change _ ≫ _ ≫ _ = _ ≫ _
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
set h := _
change IsLocalRingHom h
suffices : IsLocalRingHom ((PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _).comp h)
· apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _)
change IsLocalRingHom (_ ≫ PresheafedSpace.stalkMap (coequalizerCofork f g).π.val y)
erw [← PresheafedSpace.stalkMap.comp]
apply isLocalRingHom_stalkMap_congr _ _ (coequalizer.π_desc s.π.1 e).symm y
infer_instance
constructor
· exact LocallyRingedSpace.Hom.ext _ _ (coequalizer.π_desc _ _)
intro m h
replace h : (coequalizerCofork f g).π.1 ≫ m.1 = s.π.1 := by rw [← h]; rfl
apply LocallyRingedSpace.Hom.ext
apply (colimit.isColimit (parallelPair f.1 g.1)).uniq (Cofork.ofπ s.π.1 e) m.1
rintro ⟨⟩
· rw [← (colimit.cocone (parallelPair f.val g.val)).w WalkingParallelPairHom.left,
Category.assoc]
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case create.refine_2.right.val.zero
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
m :
((Functor.const WalkingParallelPair).obj (coequalizerCofork f g).pt).obj WalkingParallelPair.one ⟶
((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.one
h : (Cofork.π (coequalizerCofork f g)).val ≫ m.val = (Cofork.π s).val
⊢ (parallelPair f.val g.val).map WalkingParallelPairHom.left ≫
(colimit.cocone (parallelPair f.val g.val)).ι.app WalkingParallelPair.one ≫ m.val =
f.val ≫ (Cofork.π s).val
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]
apply RingHom.isUnit_map
rw [← isUnit_map_iff ((coequalizer.π f.val g.val : _).c.app _), ← comp_apply,
NatTrans.naturality, comp_apply, TopCat.Presheaf.pushforwardObj_map, ←
isUnit_map_iff (Y.presheaf.map (eqToHom hV').op)]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← comp_apply, ← Y.presheaf.map_comp]
convert @RingedSpace.isUnit_res_basicOpen Y.toRingedSpace (unop _)
(((coequalizer.π f.val g.val).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_stalk_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_stalk_isLocalRingHom
end HasCoequalizer
/-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. -/
noncomputable def coequalizer : LocallyRingedSpace where
toSheafedSpace := Limits.coequalizer f.1 g.1
localRing x := by
obtain ⟨y, rfl⟩ :=
(TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x
exact (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) y).domain_localRing
#align algebraic_geometry.LocallyRingedSpace.coequalizer AlgebraicGeometry.LocallyRingedSpace.coequalizer
/-- The explicit coequalizer cofork of locally ringed spaces. -/
noncomputable def coequalizerCofork : Cofork f g :=
@Cofork.ofπ _ _ _ _ f g (coequalizer f g) ⟨coequalizer.π f.1 g.1,
-- Porting note : this used to be automatic
HasCoequalizer.coequalizer_π_stalk_isLocalRingHom _ _⟩
(LocallyRingedSpace.Hom.ext _ _ (coequalizer.condition f.1 g.1))
#align algebraic_geometry.LocallyRingedSpace.coequalizer_cofork AlgebraicGeometry.LocallyRingedSpace.coequalizerCofork
theorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
(h : IsLocalRingHom (PresheafedSpace.stalkMap f x)) :
IsLocalRingHom (PresheafedSpace.stalkMap g x) := by
rw [PresheafedSpace.stalkMap.congr_hom _ _ H.symm x]; infer_instance
#align algebraic_geometry.LocallyRingedSpace.is_local_ring_hom_stalk_map_congr AlgebraicGeometry.LocallyRingedSpace.isLocalRingHom_stalkMap_congr
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
set h := _
change IsLocalRingHom h
suffices : IsLocalRingHom ((PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _).comp h)
· apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _)
change IsLocalRingHom (_ ≫ PresheafedSpace.stalkMap (coequalizerCofork f g).π.val y)
erw [← PresheafedSpace.stalkMap.comp]
apply isLocalRingHom_stalkMap_congr _ _ (coequalizer.π_desc s.π.1 e).symm y
infer_instance
constructor
· exact LocallyRingedSpace.Hom.ext _ _ (coequalizer.π_desc _ _)
intro m h
replace h : (coequalizerCofork f g).π.1 ≫ m.1 = s.π.1 := by rw [← h]; rfl
apply LocallyRingedSpace.Hom.ext
apply (colimit.isColimit (parallelPair f.1 g.1)).uniq (Cofork.ofπ s.π.1 e) m.1
rintro ⟨⟩
· rw [← (colimit.cocone (parallelPair f.val g.val)).w WalkingParallelPairHom.left,
Category.assoc]
change _ ≫ _ ≫ _ = _ ≫ _
|
congr
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
set h := _
change IsLocalRingHom h
suffices : IsLocalRingHom ((PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _).comp h)
· apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _)
change IsLocalRingHom (_ ≫ PresheafedSpace.stalkMap (coequalizerCofork f g).π.val y)
erw [← PresheafedSpace.stalkMap.comp]
apply isLocalRingHom_stalkMap_congr _ _ (coequalizer.π_desc s.π.1 e).symm y
infer_instance
constructor
· exact LocallyRingedSpace.Hom.ext _ _ (coequalizer.π_desc _ _)
intro m h
replace h : (coequalizerCofork f g).π.1 ≫ m.1 = s.π.1 := by rw [← h]; rfl
apply LocallyRingedSpace.Hom.ext
apply (colimit.isColimit (parallelPair f.1 g.1)).uniq (Cofork.ofπ s.π.1 e) m.1
rintro ⟨⟩
· rw [← (colimit.cocone (parallelPair f.val g.val)).w WalkingParallelPairHom.left,
Category.assoc]
change _ ≫ _ ≫ _ = _ ≫ _
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case create.refine_2.right.val.one
X Y : LocallyRingedSpace
f g : X ⟶ Y
s : Cofork f g
e : f.val ≫ (Cofork.π s).val = g.val ≫ (Cofork.π s).val
m :
((Functor.const WalkingParallelPair).obj (coequalizerCofork f g).pt).obj WalkingParallelPair.one ⟶
((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.one
h : (Cofork.π (coequalizerCofork f g)).val ≫ m.val = (Cofork.π s).val
⊢ (colimit.cocone (parallelPair f.val g.val)).ι.app WalkingParallelPair.one ≫ m.val =
(Cofork.ofπ (Cofork.π s).val e).ι.app WalkingParallelPair.one
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]
apply RingHom.isUnit_map
rw [← isUnit_map_iff ((coequalizer.π f.val g.val : _).c.app _), ← comp_apply,
NatTrans.naturality, comp_apply, TopCat.Presheaf.pushforwardObj_map, ←
isUnit_map_iff (Y.presheaf.map (eqToHom hV').op)]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← comp_apply, ← Y.presheaf.map_comp]
convert @RingedSpace.isUnit_res_basicOpen Y.toRingedSpace (unop _)
(((coequalizer.π f.val g.val).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_stalk_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_stalk_isLocalRingHom
end HasCoequalizer
/-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. -/
noncomputable def coequalizer : LocallyRingedSpace where
toSheafedSpace := Limits.coequalizer f.1 g.1
localRing x := by
obtain ⟨y, rfl⟩ :=
(TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x
exact (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) y).domain_localRing
#align algebraic_geometry.LocallyRingedSpace.coequalizer AlgebraicGeometry.LocallyRingedSpace.coequalizer
/-- The explicit coequalizer cofork of locally ringed spaces. -/
noncomputable def coequalizerCofork : Cofork f g :=
@Cofork.ofπ _ _ _ _ f g (coequalizer f g) ⟨coequalizer.π f.1 g.1,
-- Porting note : this used to be automatic
HasCoequalizer.coequalizer_π_stalk_isLocalRingHom _ _⟩
(LocallyRingedSpace.Hom.ext _ _ (coequalizer.condition f.1 g.1))
#align algebraic_geometry.LocallyRingedSpace.coequalizer_cofork AlgebraicGeometry.LocallyRingedSpace.coequalizerCofork
theorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
(h : IsLocalRingHom (PresheafedSpace.stalkMap f x)) :
IsLocalRingHom (PresheafedSpace.stalkMap g x) := by
rw [PresheafedSpace.stalkMap.congr_hom _ _ H.symm x]; infer_instance
#align algebraic_geometry.LocallyRingedSpace.is_local_ring_hom_stalk_map_congr AlgebraicGeometry.LocallyRingedSpace.isLocalRingHom_stalkMap_congr
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
set h := _
change IsLocalRingHom h
suffices : IsLocalRingHom ((PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _).comp h)
· apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _)
change IsLocalRingHom (_ ≫ PresheafedSpace.stalkMap (coequalizerCofork f g).π.val y)
erw [← PresheafedSpace.stalkMap.comp]
apply isLocalRingHom_stalkMap_congr _ _ (coequalizer.π_desc s.π.1 e).symm y
infer_instance
constructor
· exact LocallyRingedSpace.Hom.ext _ _ (coequalizer.π_desc _ _)
intro m h
replace h : (coequalizerCofork f g).π.1 ≫ m.1 = s.π.1 := by rw [← h]; rfl
apply LocallyRingedSpace.Hom.ext
apply (colimit.isColimit (parallelPair f.1 g.1)).uniq (Cofork.ofπ s.π.1 e) m.1
rintro ⟨⟩
· rw [← (colimit.cocone (parallelPair f.val g.val)).w WalkingParallelPairHom.left,
Category.assoc]
change _ ≫ _ ≫ _ = _ ≫ _
congr
·
|
exact h
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
set h := _
change IsLocalRingHom h
suffices : IsLocalRingHom ((PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _).comp h)
· apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _)
change IsLocalRingHom (_ ≫ PresheafedSpace.stalkMap (coequalizerCofork f g).π.val y)
erw [← PresheafedSpace.stalkMap.comp]
apply isLocalRingHom_stalkMap_congr _ _ (coequalizer.π_desc s.π.1 e).symm y
infer_instance
constructor
· exact LocallyRingedSpace.Hom.ext _ _ (coequalizer.π_desc _ _)
intro m h
replace h : (coequalizerCofork f g).π.1 ≫ m.1 = s.π.1 := by rw [← h]; rfl
apply LocallyRingedSpace.Hom.ext
apply (colimit.isColimit (parallelPair f.1 g.1)).uniq (Cofork.ofπ s.π.1 e) m.1
rintro ⟨⟩
· rw [← (colimit.cocone (parallelPair f.val g.val)).w WalkingParallelPairHom.left,
Category.assoc]
change _ ≫ _ ≫ _ = _ ≫ _
congr
·
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.284_0.tE6q65npbp8AX2g
|
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g)
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
X Y : LocallyRingedSpace
f g : X ⟶ Y
F : WalkingParallelPair ⥤ LocallyRingedSpace
⊢ PreservesColimit F forgetToSheafedSpace
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]
apply RingHom.isUnit_map
rw [← isUnit_map_iff ((coequalizer.π f.val g.val : _).c.app _), ← comp_apply,
NatTrans.naturality, comp_apply, TopCat.Presheaf.pushforwardObj_map, ←
isUnit_map_iff (Y.presheaf.map (eqToHom hV').op)]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← comp_apply, ← Y.presheaf.map_comp]
convert @RingedSpace.isUnit_res_basicOpen Y.toRingedSpace (unop _)
(((coequalizer.π f.val g.val).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_stalk_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_stalk_isLocalRingHom
end HasCoequalizer
/-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. -/
noncomputable def coequalizer : LocallyRingedSpace where
toSheafedSpace := Limits.coequalizer f.1 g.1
localRing x := by
obtain ⟨y, rfl⟩ :=
(TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x
exact (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) y).domain_localRing
#align algebraic_geometry.LocallyRingedSpace.coequalizer AlgebraicGeometry.LocallyRingedSpace.coequalizer
/-- The explicit coequalizer cofork of locally ringed spaces. -/
noncomputable def coequalizerCofork : Cofork f g :=
@Cofork.ofπ _ _ _ _ f g (coequalizer f g) ⟨coequalizer.π f.1 g.1,
-- Porting note : this used to be automatic
HasCoequalizer.coequalizer_π_stalk_isLocalRingHom _ _⟩
(LocallyRingedSpace.Hom.ext _ _ (coequalizer.condition f.1 g.1))
#align algebraic_geometry.LocallyRingedSpace.coequalizer_cofork AlgebraicGeometry.LocallyRingedSpace.coequalizerCofork
theorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
(h : IsLocalRingHom (PresheafedSpace.stalkMap f x)) :
IsLocalRingHom (PresheafedSpace.stalkMap g x) := by
rw [PresheafedSpace.stalkMap.congr_hom _ _ H.symm x]; infer_instance
#align algebraic_geometry.LocallyRingedSpace.is_local_ring_hom_stalk_map_congr AlgebraicGeometry.LocallyRingedSpace.isLocalRingHom_stalkMap_congr
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
set h := _
change IsLocalRingHom h
suffices : IsLocalRingHom ((PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _).comp h)
· apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _)
change IsLocalRingHom (_ ≫ PresheafedSpace.stalkMap (coequalizerCofork f g).π.val y)
erw [← PresheafedSpace.stalkMap.comp]
apply isLocalRingHom_stalkMap_congr _ _ (coequalizer.π_desc s.π.1 e).symm y
infer_instance
constructor
· exact LocallyRingedSpace.Hom.ext _ _ (coequalizer.π_desc _ _)
intro m h
replace h : (coequalizerCofork f g).π.1 ≫ m.1 = s.π.1 := by rw [← h]; rfl
apply LocallyRingedSpace.Hom.ext
apply (colimit.isColimit (parallelPair f.1 g.1)).uniq (Cofork.ofπ s.π.1 e) m.1
rintro ⟨⟩
· rw [← (colimit.cocone (parallelPair f.val g.val)).w WalkingParallelPairHom.left,
Category.assoc]
change _ ≫ _ ≫ _ = _ ≫ _
congr
· exact h
#align algebraic_geometry.LocallyRingedSpace.coequalizer_cofork_is_colimit AlgebraicGeometry.LocallyRingedSpace.coequalizerCoforkIsColimit
instance : HasCoequalizer f g :=
⟨⟨⟨_, coequalizerCoforkIsColimit f g⟩⟩⟩
instance : HasCoequalizers LocallyRingedSpace :=
hasCoequalizers_of_hasColimit_parallelPair _
noncomputable instance preservesCoequalizer :
PreservesColimitsOfShape WalkingParallelPair forgetToSheafedSpace.{v} :=
⟨fun {F} => by
-- Porting note : was `apply preservesColimitOfIsoDiagram ...` and the proof that preservation
-- of colimit is provided later
|
suffices : PreservesColimit (parallelPair (F.map WalkingParallelPairHom.left)
(F.map WalkingParallelPairHom.right)) forgetToSheafedSpace
|
noncomputable instance preservesCoequalizer :
PreservesColimitsOfShape WalkingParallelPair forgetToSheafedSpace.{v} :=
⟨fun {F} => by
-- Porting note : was `apply preservesColimitOfIsoDiagram ...` and the proof that preservation
-- of colimit is provided later
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.324_0.tE6q65npbp8AX2g
|
noncomputable instance preservesCoequalizer :
PreservesColimitsOfShape WalkingParallelPair forgetToSheafedSpace.{v}
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
X Y : LocallyRingedSpace
f g : X ⟶ Y
F : WalkingParallelPair ⥤ LocallyRingedSpace
this :
PreservesColimit (parallelPair (F.map WalkingParallelPairHom.left) (F.map WalkingParallelPairHom.right))
forgetToSheafedSpace
⊢ PreservesColimit F forgetToSheafedSpace
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]
apply RingHom.isUnit_map
rw [← isUnit_map_iff ((coequalizer.π f.val g.val : _).c.app _), ← comp_apply,
NatTrans.naturality, comp_apply, TopCat.Presheaf.pushforwardObj_map, ←
isUnit_map_iff (Y.presheaf.map (eqToHom hV').op)]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← comp_apply, ← Y.presheaf.map_comp]
convert @RingedSpace.isUnit_res_basicOpen Y.toRingedSpace (unop _)
(((coequalizer.π f.val g.val).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_stalk_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_stalk_isLocalRingHom
end HasCoequalizer
/-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. -/
noncomputable def coequalizer : LocallyRingedSpace where
toSheafedSpace := Limits.coequalizer f.1 g.1
localRing x := by
obtain ⟨y, rfl⟩ :=
(TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x
exact (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) y).domain_localRing
#align algebraic_geometry.LocallyRingedSpace.coequalizer AlgebraicGeometry.LocallyRingedSpace.coequalizer
/-- The explicit coequalizer cofork of locally ringed spaces. -/
noncomputable def coequalizerCofork : Cofork f g :=
@Cofork.ofπ _ _ _ _ f g (coequalizer f g) ⟨coequalizer.π f.1 g.1,
-- Porting note : this used to be automatic
HasCoequalizer.coequalizer_π_stalk_isLocalRingHom _ _⟩
(LocallyRingedSpace.Hom.ext _ _ (coequalizer.condition f.1 g.1))
#align algebraic_geometry.LocallyRingedSpace.coequalizer_cofork AlgebraicGeometry.LocallyRingedSpace.coequalizerCofork
theorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
(h : IsLocalRingHom (PresheafedSpace.stalkMap f x)) :
IsLocalRingHom (PresheafedSpace.stalkMap g x) := by
rw [PresheafedSpace.stalkMap.congr_hom _ _ H.symm x]; infer_instance
#align algebraic_geometry.LocallyRingedSpace.is_local_ring_hom_stalk_map_congr AlgebraicGeometry.LocallyRingedSpace.isLocalRingHom_stalkMap_congr
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
set h := _
change IsLocalRingHom h
suffices : IsLocalRingHom ((PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _).comp h)
· apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _)
change IsLocalRingHom (_ ≫ PresheafedSpace.stalkMap (coequalizerCofork f g).π.val y)
erw [← PresheafedSpace.stalkMap.comp]
apply isLocalRingHom_stalkMap_congr _ _ (coequalizer.π_desc s.π.1 e).symm y
infer_instance
constructor
· exact LocallyRingedSpace.Hom.ext _ _ (coequalizer.π_desc _ _)
intro m h
replace h : (coequalizerCofork f g).π.1 ≫ m.1 = s.π.1 := by rw [← h]; rfl
apply LocallyRingedSpace.Hom.ext
apply (colimit.isColimit (parallelPair f.1 g.1)).uniq (Cofork.ofπ s.π.1 e) m.1
rintro ⟨⟩
· rw [← (colimit.cocone (parallelPair f.val g.val)).w WalkingParallelPairHom.left,
Category.assoc]
change _ ≫ _ ≫ _ = _ ≫ _
congr
· exact h
#align algebraic_geometry.LocallyRingedSpace.coequalizer_cofork_is_colimit AlgebraicGeometry.LocallyRingedSpace.coequalizerCoforkIsColimit
instance : HasCoequalizer f g :=
⟨⟨⟨_, coequalizerCoforkIsColimit f g⟩⟩⟩
instance : HasCoequalizers LocallyRingedSpace :=
hasCoequalizers_of_hasColimit_parallelPair _
noncomputable instance preservesCoequalizer :
PreservesColimitsOfShape WalkingParallelPair forgetToSheafedSpace.{v} :=
⟨fun {F} => by
-- Porting note : was `apply preservesColimitOfIsoDiagram ...` and the proof that preservation
-- of colimit is provided later
suffices : PreservesColimit (parallelPair (F.map WalkingParallelPairHom.left)
(F.map WalkingParallelPairHom.right)) forgetToSheafedSpace
·
|
apply preservesColimitOfIsoDiagram _ (diagramIsoParallelPair F).symm
|
noncomputable instance preservesCoequalizer :
PreservesColimitsOfShape WalkingParallelPair forgetToSheafedSpace.{v} :=
⟨fun {F} => by
-- Porting note : was `apply preservesColimitOfIsoDiagram ...` and the proof that preservation
-- of colimit is provided later
suffices : PreservesColimit (parallelPair (F.map WalkingParallelPairHom.left)
(F.map WalkingParallelPairHom.right)) forgetToSheafedSpace
·
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.324_0.tE6q65npbp8AX2g
|
noncomputable instance preservesCoequalizer :
PreservesColimitsOfShape WalkingParallelPair forgetToSheafedSpace.{v}
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case this
X Y : LocallyRingedSpace
f g : X ⟶ Y
F : WalkingParallelPair ⥤ LocallyRingedSpace
⊢ PreservesColimit (parallelPair (F.map WalkingParallelPairHom.left) (F.map WalkingParallelPairHom.right))
forgetToSheafedSpace
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]
apply RingHom.isUnit_map
rw [← isUnit_map_iff ((coequalizer.π f.val g.val : _).c.app _), ← comp_apply,
NatTrans.naturality, comp_apply, TopCat.Presheaf.pushforwardObj_map, ←
isUnit_map_iff (Y.presheaf.map (eqToHom hV').op)]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← comp_apply, ← Y.presheaf.map_comp]
convert @RingedSpace.isUnit_res_basicOpen Y.toRingedSpace (unop _)
(((coequalizer.π f.val g.val).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_stalk_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_stalk_isLocalRingHom
end HasCoequalizer
/-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. -/
noncomputable def coequalizer : LocallyRingedSpace where
toSheafedSpace := Limits.coequalizer f.1 g.1
localRing x := by
obtain ⟨y, rfl⟩ :=
(TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x
exact (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) y).domain_localRing
#align algebraic_geometry.LocallyRingedSpace.coequalizer AlgebraicGeometry.LocallyRingedSpace.coequalizer
/-- The explicit coequalizer cofork of locally ringed spaces. -/
noncomputable def coequalizerCofork : Cofork f g :=
@Cofork.ofπ _ _ _ _ f g (coequalizer f g) ⟨coequalizer.π f.1 g.1,
-- Porting note : this used to be automatic
HasCoequalizer.coequalizer_π_stalk_isLocalRingHom _ _⟩
(LocallyRingedSpace.Hom.ext _ _ (coequalizer.condition f.1 g.1))
#align algebraic_geometry.LocallyRingedSpace.coequalizer_cofork AlgebraicGeometry.LocallyRingedSpace.coequalizerCofork
theorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
(h : IsLocalRingHom (PresheafedSpace.stalkMap f x)) :
IsLocalRingHom (PresheafedSpace.stalkMap g x) := by
rw [PresheafedSpace.stalkMap.congr_hom _ _ H.symm x]; infer_instance
#align algebraic_geometry.LocallyRingedSpace.is_local_ring_hom_stalk_map_congr AlgebraicGeometry.LocallyRingedSpace.isLocalRingHom_stalkMap_congr
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
set h := _
change IsLocalRingHom h
suffices : IsLocalRingHom ((PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _).comp h)
· apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _)
change IsLocalRingHom (_ ≫ PresheafedSpace.stalkMap (coequalizerCofork f g).π.val y)
erw [← PresheafedSpace.stalkMap.comp]
apply isLocalRingHom_stalkMap_congr _ _ (coequalizer.π_desc s.π.1 e).symm y
infer_instance
constructor
· exact LocallyRingedSpace.Hom.ext _ _ (coequalizer.π_desc _ _)
intro m h
replace h : (coequalizerCofork f g).π.1 ≫ m.1 = s.π.1 := by rw [← h]; rfl
apply LocallyRingedSpace.Hom.ext
apply (colimit.isColimit (parallelPair f.1 g.1)).uniq (Cofork.ofπ s.π.1 e) m.1
rintro ⟨⟩
· rw [← (colimit.cocone (parallelPair f.val g.val)).w WalkingParallelPairHom.left,
Category.assoc]
change _ ≫ _ ≫ _ = _ ≫ _
congr
· exact h
#align algebraic_geometry.LocallyRingedSpace.coequalizer_cofork_is_colimit AlgebraicGeometry.LocallyRingedSpace.coequalizerCoforkIsColimit
instance : HasCoequalizer f g :=
⟨⟨⟨_, coequalizerCoforkIsColimit f g⟩⟩⟩
instance : HasCoequalizers LocallyRingedSpace :=
hasCoequalizers_of_hasColimit_parallelPair _
noncomputable instance preservesCoequalizer :
PreservesColimitsOfShape WalkingParallelPair forgetToSheafedSpace.{v} :=
⟨fun {F} => by
-- Porting note : was `apply preservesColimitOfIsoDiagram ...` and the proof that preservation
-- of colimit is provided later
suffices : PreservesColimit (parallelPair (F.map WalkingParallelPairHom.left)
(F.map WalkingParallelPairHom.right)) forgetToSheafedSpace
· apply preservesColimitOfIsoDiagram _ (diagramIsoParallelPair F).symm
|
apply preservesColimitOfPreservesColimitCocone (coequalizerCoforkIsColimit _ _)
|
noncomputable instance preservesCoequalizer :
PreservesColimitsOfShape WalkingParallelPair forgetToSheafedSpace.{v} :=
⟨fun {F} => by
-- Porting note : was `apply preservesColimitOfIsoDiagram ...` and the proof that preservation
-- of colimit is provided later
suffices : PreservesColimit (parallelPair (F.map WalkingParallelPairHom.left)
(F.map WalkingParallelPairHom.right)) forgetToSheafedSpace
· apply preservesColimitOfIsoDiagram _ (diagramIsoParallelPair F).symm
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.324_0.tE6q65npbp8AX2g
|
noncomputable instance preservesCoequalizer :
PreservesColimitsOfShape WalkingParallelPair forgetToSheafedSpace.{v}
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
case this
X Y : LocallyRingedSpace
f g : X ⟶ Y
F : WalkingParallelPair ⥤ LocallyRingedSpace
⊢ IsColimit
(forgetToSheafedSpace.mapCocone
(coequalizerCofork (F.map WalkingParallelPairHom.left) (F.map WalkingParallelPairHom.right)))
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]
apply RingHom.isUnit_map
rw [← isUnit_map_iff ((coequalizer.π f.val g.val : _).c.app _), ← comp_apply,
NatTrans.naturality, comp_apply, TopCat.Presheaf.pushforwardObj_map, ←
isUnit_map_iff (Y.presheaf.map (eqToHom hV').op)]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← comp_apply, ← Y.presheaf.map_comp]
convert @RingedSpace.isUnit_res_basicOpen Y.toRingedSpace (unop _)
(((coequalizer.π f.val g.val).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_stalk_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_stalk_isLocalRingHom
end HasCoequalizer
/-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. -/
noncomputable def coequalizer : LocallyRingedSpace where
toSheafedSpace := Limits.coequalizer f.1 g.1
localRing x := by
obtain ⟨y, rfl⟩ :=
(TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x
exact (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) y).domain_localRing
#align algebraic_geometry.LocallyRingedSpace.coequalizer AlgebraicGeometry.LocallyRingedSpace.coequalizer
/-- The explicit coequalizer cofork of locally ringed spaces. -/
noncomputable def coequalizerCofork : Cofork f g :=
@Cofork.ofπ _ _ _ _ f g (coequalizer f g) ⟨coequalizer.π f.1 g.1,
-- Porting note : this used to be automatic
HasCoequalizer.coequalizer_π_stalk_isLocalRingHom _ _⟩
(LocallyRingedSpace.Hom.ext _ _ (coequalizer.condition f.1 g.1))
#align algebraic_geometry.LocallyRingedSpace.coequalizer_cofork AlgebraicGeometry.LocallyRingedSpace.coequalizerCofork
theorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
(h : IsLocalRingHom (PresheafedSpace.stalkMap f x)) :
IsLocalRingHom (PresheafedSpace.stalkMap g x) := by
rw [PresheafedSpace.stalkMap.congr_hom _ _ H.symm x]; infer_instance
#align algebraic_geometry.LocallyRingedSpace.is_local_ring_hom_stalk_map_congr AlgebraicGeometry.LocallyRingedSpace.isLocalRingHom_stalkMap_congr
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
set h := _
change IsLocalRingHom h
suffices : IsLocalRingHom ((PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _).comp h)
· apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _)
change IsLocalRingHom (_ ≫ PresheafedSpace.stalkMap (coequalizerCofork f g).π.val y)
erw [← PresheafedSpace.stalkMap.comp]
apply isLocalRingHom_stalkMap_congr _ _ (coequalizer.π_desc s.π.1 e).symm y
infer_instance
constructor
· exact LocallyRingedSpace.Hom.ext _ _ (coequalizer.π_desc _ _)
intro m h
replace h : (coequalizerCofork f g).π.1 ≫ m.1 = s.π.1 := by rw [← h]; rfl
apply LocallyRingedSpace.Hom.ext
apply (colimit.isColimit (parallelPair f.1 g.1)).uniq (Cofork.ofπ s.π.1 e) m.1
rintro ⟨⟩
· rw [← (colimit.cocone (parallelPair f.val g.val)).w WalkingParallelPairHom.left,
Category.assoc]
change _ ≫ _ ≫ _ = _ ≫ _
congr
· exact h
#align algebraic_geometry.LocallyRingedSpace.coequalizer_cofork_is_colimit AlgebraicGeometry.LocallyRingedSpace.coequalizerCoforkIsColimit
instance : HasCoequalizer f g :=
⟨⟨⟨_, coequalizerCoforkIsColimit f g⟩⟩⟩
instance : HasCoequalizers LocallyRingedSpace :=
hasCoequalizers_of_hasColimit_parallelPair _
noncomputable instance preservesCoequalizer :
PreservesColimitsOfShape WalkingParallelPair forgetToSheafedSpace.{v} :=
⟨fun {F} => by
-- Porting note : was `apply preservesColimitOfIsoDiagram ...` and the proof that preservation
-- of colimit is provided later
suffices : PreservesColimit (parallelPair (F.map WalkingParallelPairHom.left)
(F.map WalkingParallelPairHom.right)) forgetToSheafedSpace
· apply preservesColimitOfIsoDiagram _ (diagramIsoParallelPair F).symm
apply preservesColimitOfPreservesColimitCocone (coequalizerCoforkIsColimit _ _)
|
apply (isColimitMapCoconeCoforkEquiv _ _).symm _
|
noncomputable instance preservesCoequalizer :
PreservesColimitsOfShape WalkingParallelPair forgetToSheafedSpace.{v} :=
⟨fun {F} => by
-- Porting note : was `apply preservesColimitOfIsoDiagram ...` and the proof that preservation
-- of colimit is provided later
suffices : PreservesColimit (parallelPair (F.map WalkingParallelPairHom.left)
(F.map WalkingParallelPairHom.right)) forgetToSheafedSpace
· apply preservesColimitOfIsoDiagram _ (diagramIsoParallelPair F).symm
apply preservesColimitOfPreservesColimitCocone (coequalizerCoforkIsColimit _ _)
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.324_0.tE6q65npbp8AX2g
|
noncomputable instance preservesCoequalizer :
PreservesColimitsOfShape WalkingParallelPair forgetToSheafedSpace.{v}
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
X Y : LocallyRingedSpace
f g : X ⟶ Y
F : WalkingParallelPair ⥤ LocallyRingedSpace
⊢ IsColimit
(Cofork.ofπ
(forgetToSheafedSpace.map
{ val := coequalizer.π (F.map WalkingParallelPairHom.left).val (F.map WalkingParallelPairHom.right).val,
prop :=
(_ :
∀ (x : ↑(toTopCat (F.obj WalkingParallelPair.one))),
IsLocalRingHom
(PresheafedSpace.stalkMap
(coequalizer.π (F.map WalkingParallelPairHom.left).val (F.map WalkingParallelPairHom.right).val)
x)) })
(_ :
forgetToSheafedSpace.map (F.map WalkingParallelPairHom.left) ≫
forgetToSheafedSpace.map
{ val := coequalizer.π (F.map WalkingParallelPairHom.left).val (F.map WalkingParallelPairHom.right).val,
prop :=
(_ :
∀ (x : ↑(toTopCat (F.obj WalkingParallelPair.one))),
IsLocalRingHom
(PresheafedSpace.stalkMap
(coequalizer.π (F.map WalkingParallelPairHom.left).val
(F.map WalkingParallelPairHom.right).val)
x)) } =
forgetToSheafedSpace.map (F.map WalkingParallelPairHom.right) ≫
forgetToSheafedSpace.map
{ val := coequalizer.π (F.map WalkingParallelPairHom.left).val (F.map WalkingParallelPairHom.right).val,
prop :=
(_ :
∀ (x : ↑(toTopCat (F.obj WalkingParallelPair.one))),
IsLocalRingHom
(PresheafedSpace.stalkMap
(coequalizer.π (F.map WalkingParallelPairHom.left).val
(F.map WalkingParallelPairHom.right).val)
x)) }))
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]
apply RingHom.isUnit_map
rw [← isUnit_map_iff ((coequalizer.π f.val g.val : _).c.app _), ← comp_apply,
NatTrans.naturality, comp_apply, TopCat.Presheaf.pushforwardObj_map, ←
isUnit_map_iff (Y.presheaf.map (eqToHom hV').op)]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← comp_apply, ← Y.presheaf.map_comp]
convert @RingedSpace.isUnit_res_basicOpen Y.toRingedSpace (unop _)
(((coequalizer.π f.val g.val).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_stalk_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_stalk_isLocalRingHom
end HasCoequalizer
/-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. -/
noncomputable def coequalizer : LocallyRingedSpace where
toSheafedSpace := Limits.coequalizer f.1 g.1
localRing x := by
obtain ⟨y, rfl⟩ :=
(TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x
exact (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) y).domain_localRing
#align algebraic_geometry.LocallyRingedSpace.coequalizer AlgebraicGeometry.LocallyRingedSpace.coequalizer
/-- The explicit coequalizer cofork of locally ringed spaces. -/
noncomputable def coequalizerCofork : Cofork f g :=
@Cofork.ofπ _ _ _ _ f g (coequalizer f g) ⟨coequalizer.π f.1 g.1,
-- Porting note : this used to be automatic
HasCoequalizer.coequalizer_π_stalk_isLocalRingHom _ _⟩
(LocallyRingedSpace.Hom.ext _ _ (coequalizer.condition f.1 g.1))
#align algebraic_geometry.LocallyRingedSpace.coequalizer_cofork AlgebraicGeometry.LocallyRingedSpace.coequalizerCofork
theorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
(h : IsLocalRingHom (PresheafedSpace.stalkMap f x)) :
IsLocalRingHom (PresheafedSpace.stalkMap g x) := by
rw [PresheafedSpace.stalkMap.congr_hom _ _ H.symm x]; infer_instance
#align algebraic_geometry.LocallyRingedSpace.is_local_ring_hom_stalk_map_congr AlgebraicGeometry.LocallyRingedSpace.isLocalRingHom_stalkMap_congr
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
set h := _
change IsLocalRingHom h
suffices : IsLocalRingHom ((PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _).comp h)
· apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _)
change IsLocalRingHom (_ ≫ PresheafedSpace.stalkMap (coequalizerCofork f g).π.val y)
erw [← PresheafedSpace.stalkMap.comp]
apply isLocalRingHom_stalkMap_congr _ _ (coequalizer.π_desc s.π.1 e).symm y
infer_instance
constructor
· exact LocallyRingedSpace.Hom.ext _ _ (coequalizer.π_desc _ _)
intro m h
replace h : (coequalizerCofork f g).π.1 ≫ m.1 = s.π.1 := by rw [← h]; rfl
apply LocallyRingedSpace.Hom.ext
apply (colimit.isColimit (parallelPair f.1 g.1)).uniq (Cofork.ofπ s.π.1 e) m.1
rintro ⟨⟩
· rw [← (colimit.cocone (parallelPair f.val g.val)).w WalkingParallelPairHom.left,
Category.assoc]
change _ ≫ _ ≫ _ = _ ≫ _
congr
· exact h
#align algebraic_geometry.LocallyRingedSpace.coequalizer_cofork_is_colimit AlgebraicGeometry.LocallyRingedSpace.coequalizerCoforkIsColimit
instance : HasCoequalizer f g :=
⟨⟨⟨_, coequalizerCoforkIsColimit f g⟩⟩⟩
instance : HasCoequalizers LocallyRingedSpace :=
hasCoequalizers_of_hasColimit_parallelPair _
noncomputable instance preservesCoequalizer :
PreservesColimitsOfShape WalkingParallelPair forgetToSheafedSpace.{v} :=
⟨fun {F} => by
-- Porting note : was `apply preservesColimitOfIsoDiagram ...` and the proof that preservation
-- of colimit is provided later
suffices : PreservesColimit (parallelPair (F.map WalkingParallelPairHom.left)
(F.map WalkingParallelPairHom.right)) forgetToSheafedSpace
· apply preservesColimitOfIsoDiagram _ (diagramIsoParallelPair F).symm
apply preservesColimitOfPreservesColimitCocone (coequalizerCoforkIsColimit _ _)
apply (isColimitMapCoconeCoforkEquiv _ _).symm _
|
dsimp only [forgetToSheafedSpace]
|
noncomputable instance preservesCoequalizer :
PreservesColimitsOfShape WalkingParallelPair forgetToSheafedSpace.{v} :=
⟨fun {F} => by
-- Porting note : was `apply preservesColimitOfIsoDiagram ...` and the proof that preservation
-- of colimit is provided later
suffices : PreservesColimit (parallelPair (F.map WalkingParallelPairHom.left)
(F.map WalkingParallelPairHom.right)) forgetToSheafedSpace
· apply preservesColimitOfIsoDiagram _ (diagramIsoParallelPair F).symm
apply preservesColimitOfPreservesColimitCocone (coequalizerCoforkIsColimit _ _)
apply (isColimitMapCoconeCoforkEquiv _ _).symm _
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.324_0.tE6q65npbp8AX2g
|
noncomputable instance preservesCoequalizer :
PreservesColimitsOfShape WalkingParallelPair forgetToSheafedSpace.{v}
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
X Y : LocallyRingedSpace
f g : X ⟶ Y
F : WalkingParallelPair ⥤ LocallyRingedSpace
⊢ IsColimit
(Cofork.ofπ (coequalizer.π (F.map WalkingParallelPairHom.left).val (F.map WalkingParallelPairHom.right).val)
(_ :
(F.map WalkingParallelPairHom.left).val ≫
coequalizer.π (F.map WalkingParallelPairHom.left).val (F.map WalkingParallelPairHom.right).val =
(F.map WalkingParallelPairHom.right).val ≫
coequalizer.π (F.map WalkingParallelPairHom.left).val (F.map WalkingParallelPairHom.right).val))
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note : this instance has to be manually added
haveI : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom.c :=
PresheafedSpace.c_isIso_of_iso _
infer_instance
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom
/-!
We roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms
of locally ringed spaces, we want to show that the stalk map of
`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that
`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of
locally ringed space.
Given a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show
that `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the
restriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the
basic open set of `π⋆x`.
Since `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have
`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).
This shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`
are local ring homs.
-/
variable (U : Opens (coequalizer f.1 g.1).carrier)
variable (s : (coequalizer f.1 g.1).presheaf.obj (op U))
/-- (Implementation). The basic open set of the section `π꙳ s`. -/
noncomputable def imageBasicOpen : Opens Y :=
Y.toRingedSpace.basicOpen
(show Y.presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen
theorem imageBasicOpen_image_preimage :
(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =
(imageBasicOpen f g U s).1 := by
fapply Types.coequalizer_preimage_image_eq_of_preimage_eq
-- Porting note : Type of `f.1.base` and `g.1.base` needs to be explicit
(f.1.base : X.carrier.1 ⟶ Y.carrier.1) (g.1.base : X.carrier.1 ⟶ Y.carrier.1)
· ext
simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]
congr 2
exact coequalizer.condition f.1 g.1
· apply isColimitCoforkMapOfIsColimit (forget TopCat)
apply isColimitCoforkMapOfIsColimit (SheafedSpace.forget _)
exact coequalizerIsCoequalizer f.1 g.1
· suffices
(TopologicalSpace.Opens.map f.1.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.1.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',
SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply,
X.toRingedSpace.basicOpen_res]
apply inf_eq_right.mpr
refine' (RingedSpace.basicOpen_le _ _).trans _
rw [coequalizer.condition f.1 g.1]
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage
theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note : change `rw` to `erw`
erw [imageBasicOpen_image_preimage]
exact (imageBasicOpen f g U s).2
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open
instance coequalizer_π_stalk_isLocalRingHom (x : Y) :
IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) := by
constructor
rintro a ha
rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
erw [PresheafedSpace.stalkMap_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha
let V := imageBasicOpen f g U s
have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=
imageBasicOpen_image_preimage f g U s
have hV' :
V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=
SetLike.ext' hV.symm
have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=
imageBasicOpen_image_open f g U s
have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=
Set.image_subset_iff.mpr (Y.toRingedSpace.basicOpen_le _)
have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩
erw [←
(coequalizer f.val g.val).presheaf.germ_res_apply (homOfLE VleU)
⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]
apply RingHom.isUnit_map
rw [← isUnit_map_iff ((coequalizer.π f.val g.val : _).c.app _), ← comp_apply,
NatTrans.naturality, comp_apply, TopCat.Presheaf.pushforwardObj_map, ←
isUnit_map_iff (Y.presheaf.map (eqToHom hV').op)]
-- Porting note : change `rw` to `erw`
erw [← comp_apply, ← comp_apply, ← Y.presheaf.map_comp]
convert @RingedSpace.isUnit_res_basicOpen Y.toRingedSpace (unop _)
(((coequalizer.π f.val g.val).c.app (op U)) s)
#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_stalk_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_stalk_isLocalRingHom
end HasCoequalizer
/-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally
ringed space. -/
noncomputable def coequalizer : LocallyRingedSpace where
toSheafedSpace := Limits.coequalizer f.1 g.1
localRing x := by
obtain ⟨y, rfl⟩ :=
(TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x
exact (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) y).domain_localRing
#align algebraic_geometry.LocallyRingedSpace.coequalizer AlgebraicGeometry.LocallyRingedSpace.coequalizer
/-- The explicit coequalizer cofork of locally ringed spaces. -/
noncomputable def coequalizerCofork : Cofork f g :=
@Cofork.ofπ _ _ _ _ f g (coequalizer f g) ⟨coequalizer.π f.1 g.1,
-- Porting note : this used to be automatic
HasCoequalizer.coequalizer_π_stalk_isLocalRingHom _ _⟩
(LocallyRingedSpace.Hom.ext _ _ (coequalizer.condition f.1 g.1))
#align algebraic_geometry.LocallyRingedSpace.coequalizer_cofork AlgebraicGeometry.LocallyRingedSpace.coequalizerCofork
theorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)
(h : IsLocalRingHom (PresheafedSpace.stalkMap f x)) :
IsLocalRingHom (PresheafedSpace.stalkMap g x) := by
rw [PresheafedSpace.stalkMap.congr_hom _ _ H.symm x]; infer_instance
#align algebraic_geometry.LocallyRingedSpace.is_local_ring_hom_stalk_map_congr AlgebraicGeometry.LocallyRingedSpace.isLocalRingHom_stalkMap_congr
/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/
noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) := by
apply Cofork.IsColimit.mk'
intro s
have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition
refine ⟨⟨coequalizer.desc s.π.1 e, ?_⟩, ?_⟩
· intro x
rcases (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with
⟨y, rfl⟩
-- Porting note : was `apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap ...)`, this
-- used to allow you to provide the proof that `... ≫ ...` is a local ring homomorphism later,
-- but this is no longer possible
set h := _
change IsLocalRingHom h
suffices : IsLocalRingHom ((PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _).comp h)
· apply isLocalRingHom_of_comp _ (PresheafedSpace.stalkMap (coequalizerCofork f g).π.1 _)
change IsLocalRingHom (_ ≫ PresheafedSpace.stalkMap (coequalizerCofork f g).π.val y)
erw [← PresheafedSpace.stalkMap.comp]
apply isLocalRingHom_stalkMap_congr _ _ (coequalizer.π_desc s.π.1 e).symm y
infer_instance
constructor
· exact LocallyRingedSpace.Hom.ext _ _ (coequalizer.π_desc _ _)
intro m h
replace h : (coequalizerCofork f g).π.1 ≫ m.1 = s.π.1 := by rw [← h]; rfl
apply LocallyRingedSpace.Hom.ext
apply (colimit.isColimit (parallelPair f.1 g.1)).uniq (Cofork.ofπ s.π.1 e) m.1
rintro ⟨⟩
· rw [← (colimit.cocone (parallelPair f.val g.val)).w WalkingParallelPairHom.left,
Category.assoc]
change _ ≫ _ ≫ _ = _ ≫ _
congr
· exact h
#align algebraic_geometry.LocallyRingedSpace.coequalizer_cofork_is_colimit AlgebraicGeometry.LocallyRingedSpace.coequalizerCoforkIsColimit
instance : HasCoequalizer f g :=
⟨⟨⟨_, coequalizerCoforkIsColimit f g⟩⟩⟩
instance : HasCoequalizers LocallyRingedSpace :=
hasCoequalizers_of_hasColimit_parallelPair _
noncomputable instance preservesCoequalizer :
PreservesColimitsOfShape WalkingParallelPair forgetToSheafedSpace.{v} :=
⟨fun {F} => by
-- Porting note : was `apply preservesColimitOfIsoDiagram ...` and the proof that preservation
-- of colimit is provided later
suffices : PreservesColimit (parallelPair (F.map WalkingParallelPairHom.left)
(F.map WalkingParallelPairHom.right)) forgetToSheafedSpace
· apply preservesColimitOfIsoDiagram _ (diagramIsoParallelPair F).symm
apply preservesColimitOfPreservesColimitCocone (coequalizerCoforkIsColimit _ _)
apply (isColimitMapCoconeCoforkEquiv _ _).symm _
dsimp only [forgetToSheafedSpace]
|
exact coequalizerIsCoequalizer _ _
|
noncomputable instance preservesCoequalizer :
PreservesColimitsOfShape WalkingParallelPair forgetToSheafedSpace.{v} :=
⟨fun {F} => by
-- Porting note : was `apply preservesColimitOfIsoDiagram ...` and the proof that preservation
-- of colimit is provided later
suffices : PreservesColimit (parallelPair (F.map WalkingParallelPairHom.left)
(F.map WalkingParallelPairHom.right)) forgetToSheafedSpace
· apply preservesColimitOfIsoDiagram _ (diagramIsoParallelPair F).symm
apply preservesColimitOfPreservesColimitCocone (coequalizerCoforkIsColimit _ _)
apply (isColimitMapCoconeCoforkEquiv _ _).symm _
dsimp only [forgetToSheafedSpace]
|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.324_0.tE6q65npbp8AX2g
|
noncomputable instance preservesCoequalizer :
PreservesColimitsOfShape WalkingParallelPair forgetToSheafedSpace.{v}
|
Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits
|
R : Type u
inst✝ : AddGroupWithOne R
m n : ℕ
h : m ≤ n
⊢ ↑(n - m) + ↑m = ↑n
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Cast of integers (additional theorems)
This file proves additional properties about the *canonical* homomorphism from
the integers into an additive group with a one (`Int.cast`).
There is also `Data.Int.Cast.Lemmas`,
which includes lemmas stated in terms of algebraic homomorphisms,
and results involving the order structure of `ℤ`.
By contrast, this file's only import beyond `Data.Int.Cast.Defs` is `Algebra.Group.Basic`.
-/
universe u
namespace Nat
variable {R : Type u} [AddGroupWithOne R]
@[simp, norm_cast]
theorem cast_sub {m n} (h : m ≤ n) : ((n - m : ℕ) : R) = n - m :=
eq_sub_of_add_eq <| by
|
rw [← cast_add, Nat.sub_add_cancel h]
|
@[simp, norm_cast]
theorem cast_sub {m n} (h : m ≤ n) : ((n - m : ℕ) : R) = n - m :=
eq_sub_of_add_eq <| by
|
Mathlib.Data.Int.Cast.Basic.32_0.3MsWc9B5PAFbTbn
|
@[simp, norm_cast]
theorem cast_sub {m n} (h : m ≤ n) : ((n - m : ℕ) : R) = n - m
|
Mathlib_Data_Int_Cast_Basic
|
R : Type u
inst✝ : AddGroupWithOne R
h : 0 < 0
⊢ ↑(0 - 1) = ↑0 - 1
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Cast of integers (additional theorems)
This file proves additional properties about the *canonical* homomorphism from
the integers into an additive group with a one (`Int.cast`).
There is also `Data.Int.Cast.Lemmas`,
which includes lemmas stated in terms of algebraic homomorphisms,
and results involving the order structure of `ℤ`.
By contrast, this file's only import beyond `Data.Int.Cast.Defs` is `Algebra.Group.Basic`.
-/
universe u
namespace Nat
variable {R : Type u} [AddGroupWithOne R]
@[simp, norm_cast]
theorem cast_sub {m n} (h : m ≤ n) : ((n - m : ℕ) : R) = n - m :=
eq_sub_of_add_eq <| by rw [← cast_add, Nat.sub_add_cancel h]
#align nat.cast_sub Nat.cast_subₓ
-- `HasLiftT` appeared in the type signature
@[simp, norm_cast]
theorem cast_pred : ∀ {n}, 0 < n → ((n - 1 : ℕ) : R) = n - 1
| 0, h => by
|
cases h
|
@[simp, norm_cast]
theorem cast_pred : ∀ {n}, 0 < n → ((n - 1 : ℕ) : R) = n - 1
| 0, h => by
|
Mathlib.Data.Int.Cast.Basic.38_0.3MsWc9B5PAFbTbn
|
@[simp, norm_cast]
theorem cast_pred : ∀ {n}, 0 < n → ((n - 1 : ℕ) : R) = n - 1
| 0, h => by cases h
| n + 1, _ => by rw [cast_succ, add_sub_cancel]; rfl
|
Mathlib_Data_Int_Cast_Basic
|
R : Type u
inst✝ : AddGroupWithOne R
n : ℕ
x✝ : 0 < n + 1
⊢ ↑(n + 1 - 1) = ↑(n + 1) - 1
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Cast of integers (additional theorems)
This file proves additional properties about the *canonical* homomorphism from
the integers into an additive group with a one (`Int.cast`).
There is also `Data.Int.Cast.Lemmas`,
which includes lemmas stated in terms of algebraic homomorphisms,
and results involving the order structure of `ℤ`.
By contrast, this file's only import beyond `Data.Int.Cast.Defs` is `Algebra.Group.Basic`.
-/
universe u
namespace Nat
variable {R : Type u} [AddGroupWithOne R]
@[simp, norm_cast]
theorem cast_sub {m n} (h : m ≤ n) : ((n - m : ℕ) : R) = n - m :=
eq_sub_of_add_eq <| by rw [← cast_add, Nat.sub_add_cancel h]
#align nat.cast_sub Nat.cast_subₓ
-- `HasLiftT` appeared in the type signature
@[simp, norm_cast]
theorem cast_pred : ∀ {n}, 0 < n → ((n - 1 : ℕ) : R) = n - 1
| 0, h => by cases h
| n + 1, _ => by
|
rw [cast_succ, add_sub_cancel]
|
@[simp, norm_cast]
theorem cast_pred : ∀ {n}, 0 < n → ((n - 1 : ℕ) : R) = n - 1
| 0, h => by cases h
| n + 1, _ => by
|
Mathlib.Data.Int.Cast.Basic.38_0.3MsWc9B5PAFbTbn
|
@[simp, norm_cast]
theorem cast_pred : ∀ {n}, 0 < n → ((n - 1 : ℕ) : R) = n - 1
| 0, h => by cases h
| n + 1, _ => by rw [cast_succ, add_sub_cancel]; rfl
|
Mathlib_Data_Int_Cast_Basic
|
R : Type u
inst✝ : AddGroupWithOne R
n : ℕ
x✝ : 0 < n + 1
⊢ ↑(n + 1 - 1) = ↑n
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Cast of integers (additional theorems)
This file proves additional properties about the *canonical* homomorphism from
the integers into an additive group with a one (`Int.cast`).
There is also `Data.Int.Cast.Lemmas`,
which includes lemmas stated in terms of algebraic homomorphisms,
and results involving the order structure of `ℤ`.
By contrast, this file's only import beyond `Data.Int.Cast.Defs` is `Algebra.Group.Basic`.
-/
universe u
namespace Nat
variable {R : Type u} [AddGroupWithOne R]
@[simp, norm_cast]
theorem cast_sub {m n} (h : m ≤ n) : ((n - m : ℕ) : R) = n - m :=
eq_sub_of_add_eq <| by rw [← cast_add, Nat.sub_add_cancel h]
#align nat.cast_sub Nat.cast_subₓ
-- `HasLiftT` appeared in the type signature
@[simp, norm_cast]
theorem cast_pred : ∀ {n}, 0 < n → ((n - 1 : ℕ) : R) = n - 1
| 0, h => by cases h
| n + 1, _ => by rw [cast_succ, add_sub_cancel];
|
rfl
|
@[simp, norm_cast]
theorem cast_pred : ∀ {n}, 0 < n → ((n - 1 : ℕ) : R) = n - 1
| 0, h => by cases h
| n + 1, _ => by rw [cast_succ, add_sub_cancel];
|
Mathlib.Data.Int.Cast.Basic.38_0.3MsWc9B5PAFbTbn
|
@[simp, norm_cast]
theorem cast_pred : ∀ {n}, 0 < n → ((n - 1 : ℕ) : R) = n - 1
| 0, h => by cases h
| n + 1, _ => by rw [cast_succ, add_sub_cancel]; rfl
|
Mathlib_Data_Int_Cast_Basic
|
R : Type u
inst✝¹ : AddGroupWithOne R
n : ℕ
inst✝ : AtLeastTwo n
⊢ ↑(OfNat.ofNat n) = OfNat.ofNat n
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Cast of integers (additional theorems)
This file proves additional properties about the *canonical* homomorphism from
the integers into an additive group with a one (`Int.cast`).
There is also `Data.Int.Cast.Lemmas`,
which includes lemmas stated in terms of algebraic homomorphisms,
and results involving the order structure of `ℤ`.
By contrast, this file's only import beyond `Data.Int.Cast.Defs` is `Algebra.Group.Basic`.
-/
universe u
namespace Nat
variable {R : Type u} [AddGroupWithOne R]
@[simp, norm_cast]
theorem cast_sub {m n} (h : m ≤ n) : ((n - m : ℕ) : R) = n - m :=
eq_sub_of_add_eq <| by rw [← cast_add, Nat.sub_add_cancel h]
#align nat.cast_sub Nat.cast_subₓ
-- `HasLiftT` appeared in the type signature
@[simp, norm_cast]
theorem cast_pred : ∀ {n}, 0 < n → ((n - 1 : ℕ) : R) = n - 1
| 0, h => by cases h
| n + 1, _ => by rw [cast_succ, add_sub_cancel]; rfl
#align nat.cast_pred Nat.cast_pred
end Nat
open Nat
namespace Int
variable {R : Type u} [AddGroupWithOne R]
@[simp, norm_cast squash]
theorem cast_negSucc (n : ℕ) : (-[n+1] : R) = -(n + 1 : ℕ) :=
AddGroupWithOne.intCast_negSucc n
#align int.cast_neg_succ_of_nat Int.cast_negSuccₓ
-- expected `n` to be implicit, and `HasLiftT`
@[simp, norm_cast]
theorem cast_zero : ((0 : ℤ) : R) = 0 :=
(AddGroupWithOne.intCast_ofNat 0).trans Nat.cast_zero
#align int.cast_zero Int.cast_zeroₓ
-- type had `HasLiftT`
@[simp high, nolint simpNF, norm_cast] -- this lemma competes with `Int.ofNat_eq_cast` to come later
theorem cast_ofNat (n : ℕ) : ((n : ℤ) : R) = n :=
AddGroupWithOne.intCast_ofNat _
#align int.cast_coe_nat Int.cast_ofNatₓ
-- expected `n` to be implicit, and `HasLiftT`
#align int.cast_of_nat Int.cast_ofNatₓ
-- See note [no_index around OfNat.ofNat]
@[simp, norm_cast]
theorem int_cast_ofNat (n : ℕ) [n.AtLeastTwo] :
((no_index (OfNat.ofNat n) : ℤ) : R) = OfNat.ofNat n := by
|
simpa only [OfNat.ofNat] using AddGroupWithOne.intCast_ofNat (R := R) n
|
@[simp, norm_cast]
theorem int_cast_ofNat (n : ℕ) [n.AtLeastTwo] :
((no_index (OfNat.ofNat n) : ℤ) : R) = OfNat.ofNat n := by
|
Mathlib.Data.Int.Cast.Basic.72_0.3MsWc9B5PAFbTbn
|
@[simp, norm_cast]
theorem int_cast_ofNat (n : ℕ) [n.AtLeastTwo] :
((no_index (OfNat.ofNat n) : ℤ) : R) = OfNat.ofNat n
|
Mathlib_Data_Int_Cast_Basic
|
R : Type u
inst✝ : AddGroupWithOne R
⊢ ↑1 = 1
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Cast of integers (additional theorems)
This file proves additional properties about the *canonical* homomorphism from
the integers into an additive group with a one (`Int.cast`).
There is also `Data.Int.Cast.Lemmas`,
which includes lemmas stated in terms of algebraic homomorphisms,
and results involving the order structure of `ℤ`.
By contrast, this file's only import beyond `Data.Int.Cast.Defs` is `Algebra.Group.Basic`.
-/
universe u
namespace Nat
variable {R : Type u} [AddGroupWithOne R]
@[simp, norm_cast]
theorem cast_sub {m n} (h : m ≤ n) : ((n - m : ℕ) : R) = n - m :=
eq_sub_of_add_eq <| by rw [← cast_add, Nat.sub_add_cancel h]
#align nat.cast_sub Nat.cast_subₓ
-- `HasLiftT` appeared in the type signature
@[simp, norm_cast]
theorem cast_pred : ∀ {n}, 0 < n → ((n - 1 : ℕ) : R) = n - 1
| 0, h => by cases h
| n + 1, _ => by rw [cast_succ, add_sub_cancel]; rfl
#align nat.cast_pred Nat.cast_pred
end Nat
open Nat
namespace Int
variable {R : Type u} [AddGroupWithOne R]
@[simp, norm_cast squash]
theorem cast_negSucc (n : ℕ) : (-[n+1] : R) = -(n + 1 : ℕ) :=
AddGroupWithOne.intCast_negSucc n
#align int.cast_neg_succ_of_nat Int.cast_negSuccₓ
-- expected `n` to be implicit, and `HasLiftT`
@[simp, norm_cast]
theorem cast_zero : ((0 : ℤ) : R) = 0 :=
(AddGroupWithOne.intCast_ofNat 0).trans Nat.cast_zero
#align int.cast_zero Int.cast_zeroₓ
-- type had `HasLiftT`
@[simp high, nolint simpNF, norm_cast] -- this lemma competes with `Int.ofNat_eq_cast` to come later
theorem cast_ofNat (n : ℕ) : ((n : ℤ) : R) = n :=
AddGroupWithOne.intCast_ofNat _
#align int.cast_coe_nat Int.cast_ofNatₓ
-- expected `n` to be implicit, and `HasLiftT`
#align int.cast_of_nat Int.cast_ofNatₓ
-- See note [no_index around OfNat.ofNat]
@[simp, norm_cast]
theorem int_cast_ofNat (n : ℕ) [n.AtLeastTwo] :
((no_index (OfNat.ofNat n) : ℤ) : R) = OfNat.ofNat n := by
simpa only [OfNat.ofNat] using AddGroupWithOne.intCast_ofNat (R := R) n
@[simp, norm_cast]
theorem cast_one : ((1 : ℤ) : R) = 1 := by
|
erw [cast_ofNat, Nat.cast_one]
|
@[simp, norm_cast]
theorem cast_one : ((1 : ℤ) : R) = 1 := by
|
Mathlib.Data.Int.Cast.Basic.77_0.3MsWc9B5PAFbTbn
|
@[simp, norm_cast]
theorem cast_one : ((1 : ℤ) : R) = 1
|
Mathlib_Data_Int_Cast_Basic
|
R : Type u
inst✝ : AddGroupWithOne R
⊢ ↑(-↑0) = -↑↑0
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Cast of integers (additional theorems)
This file proves additional properties about the *canonical* homomorphism from
the integers into an additive group with a one (`Int.cast`).
There is also `Data.Int.Cast.Lemmas`,
which includes lemmas stated in terms of algebraic homomorphisms,
and results involving the order structure of `ℤ`.
By contrast, this file's only import beyond `Data.Int.Cast.Defs` is `Algebra.Group.Basic`.
-/
universe u
namespace Nat
variable {R : Type u} [AddGroupWithOne R]
@[simp, norm_cast]
theorem cast_sub {m n} (h : m ≤ n) : ((n - m : ℕ) : R) = n - m :=
eq_sub_of_add_eq <| by rw [← cast_add, Nat.sub_add_cancel h]
#align nat.cast_sub Nat.cast_subₓ
-- `HasLiftT` appeared in the type signature
@[simp, norm_cast]
theorem cast_pred : ∀ {n}, 0 < n → ((n - 1 : ℕ) : R) = n - 1
| 0, h => by cases h
| n + 1, _ => by rw [cast_succ, add_sub_cancel]; rfl
#align nat.cast_pred Nat.cast_pred
end Nat
open Nat
namespace Int
variable {R : Type u} [AddGroupWithOne R]
@[simp, norm_cast squash]
theorem cast_negSucc (n : ℕ) : (-[n+1] : R) = -(n + 1 : ℕ) :=
AddGroupWithOne.intCast_negSucc n
#align int.cast_neg_succ_of_nat Int.cast_negSuccₓ
-- expected `n` to be implicit, and `HasLiftT`
@[simp, norm_cast]
theorem cast_zero : ((0 : ℤ) : R) = 0 :=
(AddGroupWithOne.intCast_ofNat 0).trans Nat.cast_zero
#align int.cast_zero Int.cast_zeroₓ
-- type had `HasLiftT`
@[simp high, nolint simpNF, norm_cast] -- this lemma competes with `Int.ofNat_eq_cast` to come later
theorem cast_ofNat (n : ℕ) : ((n : ℤ) : R) = n :=
AddGroupWithOne.intCast_ofNat _
#align int.cast_coe_nat Int.cast_ofNatₓ
-- expected `n` to be implicit, and `HasLiftT`
#align int.cast_of_nat Int.cast_ofNatₓ
-- See note [no_index around OfNat.ofNat]
@[simp, norm_cast]
theorem int_cast_ofNat (n : ℕ) [n.AtLeastTwo] :
((no_index (OfNat.ofNat n) : ℤ) : R) = OfNat.ofNat n := by
simpa only [OfNat.ofNat] using AddGroupWithOne.intCast_ofNat (R := R) n
@[simp, norm_cast]
theorem cast_one : ((1 : ℤ) : R) = 1 := by
erw [cast_ofNat, Nat.cast_one]
#align int.cast_one Int.cast_oneₓ
-- type had `HasLiftT`
@[simp, norm_cast]
theorem cast_neg : ∀ n, ((-n : ℤ) : R) = -n
| (0 : ℕ) => by
|
erw [cast_zero, neg_zero]
|
@[simp, norm_cast]
theorem cast_neg : ∀ n, ((-n : ℤ) : R) = -n
| (0 : ℕ) => by
|
Mathlib.Data.Int.Cast.Basic.83_0.3MsWc9B5PAFbTbn
|
@[simp, norm_cast]
theorem cast_neg : ∀ n, ((-n : ℤ) : R) = -n
| (0 : ℕ) => by erw [cast_zero, neg_zero]
| (n + 1 : ℕ) => by erw [cast_ofNat, cast_negSucc]
| -[n+1] => by erw [cast_ofNat, cast_negSucc, neg_neg]
|
Mathlib_Data_Int_Cast_Basic
|
R : Type u
inst✝ : AddGroupWithOne R
n : ℕ
⊢ ↑(-↑(n + 1)) = -↑↑(n + 1)
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Cast of integers (additional theorems)
This file proves additional properties about the *canonical* homomorphism from
the integers into an additive group with a one (`Int.cast`).
There is also `Data.Int.Cast.Lemmas`,
which includes lemmas stated in terms of algebraic homomorphisms,
and results involving the order structure of `ℤ`.
By contrast, this file's only import beyond `Data.Int.Cast.Defs` is `Algebra.Group.Basic`.
-/
universe u
namespace Nat
variable {R : Type u} [AddGroupWithOne R]
@[simp, norm_cast]
theorem cast_sub {m n} (h : m ≤ n) : ((n - m : ℕ) : R) = n - m :=
eq_sub_of_add_eq <| by rw [← cast_add, Nat.sub_add_cancel h]
#align nat.cast_sub Nat.cast_subₓ
-- `HasLiftT` appeared in the type signature
@[simp, norm_cast]
theorem cast_pred : ∀ {n}, 0 < n → ((n - 1 : ℕ) : R) = n - 1
| 0, h => by cases h
| n + 1, _ => by rw [cast_succ, add_sub_cancel]; rfl
#align nat.cast_pred Nat.cast_pred
end Nat
open Nat
namespace Int
variable {R : Type u} [AddGroupWithOne R]
@[simp, norm_cast squash]
theorem cast_negSucc (n : ℕ) : (-[n+1] : R) = -(n + 1 : ℕ) :=
AddGroupWithOne.intCast_negSucc n
#align int.cast_neg_succ_of_nat Int.cast_negSuccₓ
-- expected `n` to be implicit, and `HasLiftT`
@[simp, norm_cast]
theorem cast_zero : ((0 : ℤ) : R) = 0 :=
(AddGroupWithOne.intCast_ofNat 0).trans Nat.cast_zero
#align int.cast_zero Int.cast_zeroₓ
-- type had `HasLiftT`
@[simp high, nolint simpNF, norm_cast] -- this lemma competes with `Int.ofNat_eq_cast` to come later
theorem cast_ofNat (n : ℕ) : ((n : ℤ) : R) = n :=
AddGroupWithOne.intCast_ofNat _
#align int.cast_coe_nat Int.cast_ofNatₓ
-- expected `n` to be implicit, and `HasLiftT`
#align int.cast_of_nat Int.cast_ofNatₓ
-- See note [no_index around OfNat.ofNat]
@[simp, norm_cast]
theorem int_cast_ofNat (n : ℕ) [n.AtLeastTwo] :
((no_index (OfNat.ofNat n) : ℤ) : R) = OfNat.ofNat n := by
simpa only [OfNat.ofNat] using AddGroupWithOne.intCast_ofNat (R := R) n
@[simp, norm_cast]
theorem cast_one : ((1 : ℤ) : R) = 1 := by
erw [cast_ofNat, Nat.cast_one]
#align int.cast_one Int.cast_oneₓ
-- type had `HasLiftT`
@[simp, norm_cast]
theorem cast_neg : ∀ n, ((-n : ℤ) : R) = -n
| (0 : ℕ) => by erw [cast_zero, neg_zero]
| (n + 1 : ℕ) => by
|
erw [cast_ofNat, cast_negSucc]
|
@[simp, norm_cast]
theorem cast_neg : ∀ n, ((-n : ℤ) : R) = -n
| (0 : ℕ) => by erw [cast_zero, neg_zero]
| (n + 1 : ℕ) => by
|
Mathlib.Data.Int.Cast.Basic.83_0.3MsWc9B5PAFbTbn
|
@[simp, norm_cast]
theorem cast_neg : ∀ n, ((-n : ℤ) : R) = -n
| (0 : ℕ) => by erw [cast_zero, neg_zero]
| (n + 1 : ℕ) => by erw [cast_ofNat, cast_negSucc]
| -[n+1] => by erw [cast_ofNat, cast_negSucc, neg_neg]
|
Mathlib_Data_Int_Cast_Basic
|
R : Type u
inst✝ : AddGroupWithOne R
n : ℕ
⊢ ↑(- -[n+1]) = -↑-[n+1]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Cast of integers (additional theorems)
This file proves additional properties about the *canonical* homomorphism from
the integers into an additive group with a one (`Int.cast`).
There is also `Data.Int.Cast.Lemmas`,
which includes lemmas stated in terms of algebraic homomorphisms,
and results involving the order structure of `ℤ`.
By contrast, this file's only import beyond `Data.Int.Cast.Defs` is `Algebra.Group.Basic`.
-/
universe u
namespace Nat
variable {R : Type u} [AddGroupWithOne R]
@[simp, norm_cast]
theorem cast_sub {m n} (h : m ≤ n) : ((n - m : ℕ) : R) = n - m :=
eq_sub_of_add_eq <| by rw [← cast_add, Nat.sub_add_cancel h]
#align nat.cast_sub Nat.cast_subₓ
-- `HasLiftT` appeared in the type signature
@[simp, norm_cast]
theorem cast_pred : ∀ {n}, 0 < n → ((n - 1 : ℕ) : R) = n - 1
| 0, h => by cases h
| n + 1, _ => by rw [cast_succ, add_sub_cancel]; rfl
#align nat.cast_pred Nat.cast_pred
end Nat
open Nat
namespace Int
variable {R : Type u} [AddGroupWithOne R]
@[simp, norm_cast squash]
theorem cast_negSucc (n : ℕ) : (-[n+1] : R) = -(n + 1 : ℕ) :=
AddGroupWithOne.intCast_negSucc n
#align int.cast_neg_succ_of_nat Int.cast_negSuccₓ
-- expected `n` to be implicit, and `HasLiftT`
@[simp, norm_cast]
theorem cast_zero : ((0 : ℤ) : R) = 0 :=
(AddGroupWithOne.intCast_ofNat 0).trans Nat.cast_zero
#align int.cast_zero Int.cast_zeroₓ
-- type had `HasLiftT`
@[simp high, nolint simpNF, norm_cast] -- this lemma competes with `Int.ofNat_eq_cast` to come later
theorem cast_ofNat (n : ℕ) : ((n : ℤ) : R) = n :=
AddGroupWithOne.intCast_ofNat _
#align int.cast_coe_nat Int.cast_ofNatₓ
-- expected `n` to be implicit, and `HasLiftT`
#align int.cast_of_nat Int.cast_ofNatₓ
-- See note [no_index around OfNat.ofNat]
@[simp, norm_cast]
theorem int_cast_ofNat (n : ℕ) [n.AtLeastTwo] :
((no_index (OfNat.ofNat n) : ℤ) : R) = OfNat.ofNat n := by
simpa only [OfNat.ofNat] using AddGroupWithOne.intCast_ofNat (R := R) n
@[simp, norm_cast]
theorem cast_one : ((1 : ℤ) : R) = 1 := by
erw [cast_ofNat, Nat.cast_one]
#align int.cast_one Int.cast_oneₓ
-- type had `HasLiftT`
@[simp, norm_cast]
theorem cast_neg : ∀ n, ((-n : ℤ) : R) = -n
| (0 : ℕ) => by erw [cast_zero, neg_zero]
| (n + 1 : ℕ) => by erw [cast_ofNat, cast_negSucc]
| -[n+1] => by
|
erw [cast_ofNat, cast_negSucc, neg_neg]
|
@[simp, norm_cast]
theorem cast_neg : ∀ n, ((-n : ℤ) : R) = -n
| (0 : ℕ) => by erw [cast_zero, neg_zero]
| (n + 1 : ℕ) => by erw [cast_ofNat, cast_negSucc]
| -[n+1] => by
|
Mathlib.Data.Int.Cast.Basic.83_0.3MsWc9B5PAFbTbn
|
@[simp, norm_cast]
theorem cast_neg : ∀ n, ((-n : ℤ) : R) = -n
| (0 : ℕ) => by erw [cast_zero, neg_zero]
| (n + 1 : ℕ) => by erw [cast_ofNat, cast_negSucc]
| -[n+1] => by erw [cast_ofNat, cast_negSucc, neg_neg]
|
Mathlib_Data_Int_Cast_Basic
|
R : Type u
inst✝ : AddGroupWithOne R
m n : ℕ
⊢ ↑(subNatNat m n) = ↑m - ↑n
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Cast of integers (additional theorems)
This file proves additional properties about the *canonical* homomorphism from
the integers into an additive group with a one (`Int.cast`).
There is also `Data.Int.Cast.Lemmas`,
which includes lemmas stated in terms of algebraic homomorphisms,
and results involving the order structure of `ℤ`.
By contrast, this file's only import beyond `Data.Int.Cast.Defs` is `Algebra.Group.Basic`.
-/
universe u
namespace Nat
variable {R : Type u} [AddGroupWithOne R]
@[simp, norm_cast]
theorem cast_sub {m n} (h : m ≤ n) : ((n - m : ℕ) : R) = n - m :=
eq_sub_of_add_eq <| by rw [← cast_add, Nat.sub_add_cancel h]
#align nat.cast_sub Nat.cast_subₓ
-- `HasLiftT` appeared in the type signature
@[simp, norm_cast]
theorem cast_pred : ∀ {n}, 0 < n → ((n - 1 : ℕ) : R) = n - 1
| 0, h => by cases h
| n + 1, _ => by rw [cast_succ, add_sub_cancel]; rfl
#align nat.cast_pred Nat.cast_pred
end Nat
open Nat
namespace Int
variable {R : Type u} [AddGroupWithOne R]
@[simp, norm_cast squash]
theorem cast_negSucc (n : ℕ) : (-[n+1] : R) = -(n + 1 : ℕ) :=
AddGroupWithOne.intCast_negSucc n
#align int.cast_neg_succ_of_nat Int.cast_negSuccₓ
-- expected `n` to be implicit, and `HasLiftT`
@[simp, norm_cast]
theorem cast_zero : ((0 : ℤ) : R) = 0 :=
(AddGroupWithOne.intCast_ofNat 0).trans Nat.cast_zero
#align int.cast_zero Int.cast_zeroₓ
-- type had `HasLiftT`
@[simp high, nolint simpNF, norm_cast] -- this lemma competes with `Int.ofNat_eq_cast` to come later
theorem cast_ofNat (n : ℕ) : ((n : ℤ) : R) = n :=
AddGroupWithOne.intCast_ofNat _
#align int.cast_coe_nat Int.cast_ofNatₓ
-- expected `n` to be implicit, and `HasLiftT`
#align int.cast_of_nat Int.cast_ofNatₓ
-- See note [no_index around OfNat.ofNat]
@[simp, norm_cast]
theorem int_cast_ofNat (n : ℕ) [n.AtLeastTwo] :
((no_index (OfNat.ofNat n) : ℤ) : R) = OfNat.ofNat n := by
simpa only [OfNat.ofNat] using AddGroupWithOne.intCast_ofNat (R := R) n
@[simp, norm_cast]
theorem cast_one : ((1 : ℤ) : R) = 1 := by
erw [cast_ofNat, Nat.cast_one]
#align int.cast_one Int.cast_oneₓ
-- type had `HasLiftT`
@[simp, norm_cast]
theorem cast_neg : ∀ n, ((-n : ℤ) : R) = -n
| (0 : ℕ) => by erw [cast_zero, neg_zero]
| (n + 1 : ℕ) => by erw [cast_ofNat, cast_negSucc]
| -[n+1] => by erw [cast_ofNat, cast_negSucc, neg_neg]
#align int.cast_neg Int.cast_negₓ
-- type had `HasLiftT`
@[simp, norm_cast]
theorem cast_subNatNat (m n) : ((Int.subNatNat m n : ℤ) : R) = m - n := by
|
unfold subNatNat
|
@[simp, norm_cast]
theorem cast_subNatNat (m n) : ((Int.subNatNat m n : ℤ) : R) = m - n := by
|
Mathlib.Data.Int.Cast.Basic.91_0.3MsWc9B5PAFbTbn
|
@[simp, norm_cast]
theorem cast_subNatNat (m n) : ((Int.subNatNat m n : ℤ) : R) = m - n
|
Mathlib_Data_Int_Cast_Basic
|
R : Type u
inst✝ : AddGroupWithOne R
m n : ℕ
⊢ ↑(match n - m with
| 0 => ofNat (m - n)
| succ k => -[k+1]) =
↑m - ↑n
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Cast of integers (additional theorems)
This file proves additional properties about the *canonical* homomorphism from
the integers into an additive group with a one (`Int.cast`).
There is also `Data.Int.Cast.Lemmas`,
which includes lemmas stated in terms of algebraic homomorphisms,
and results involving the order structure of `ℤ`.
By contrast, this file's only import beyond `Data.Int.Cast.Defs` is `Algebra.Group.Basic`.
-/
universe u
namespace Nat
variable {R : Type u} [AddGroupWithOne R]
@[simp, norm_cast]
theorem cast_sub {m n} (h : m ≤ n) : ((n - m : ℕ) : R) = n - m :=
eq_sub_of_add_eq <| by rw [← cast_add, Nat.sub_add_cancel h]
#align nat.cast_sub Nat.cast_subₓ
-- `HasLiftT` appeared in the type signature
@[simp, norm_cast]
theorem cast_pred : ∀ {n}, 0 < n → ((n - 1 : ℕ) : R) = n - 1
| 0, h => by cases h
| n + 1, _ => by rw [cast_succ, add_sub_cancel]; rfl
#align nat.cast_pred Nat.cast_pred
end Nat
open Nat
namespace Int
variable {R : Type u} [AddGroupWithOne R]
@[simp, norm_cast squash]
theorem cast_negSucc (n : ℕ) : (-[n+1] : R) = -(n + 1 : ℕ) :=
AddGroupWithOne.intCast_negSucc n
#align int.cast_neg_succ_of_nat Int.cast_negSuccₓ
-- expected `n` to be implicit, and `HasLiftT`
@[simp, norm_cast]
theorem cast_zero : ((0 : ℤ) : R) = 0 :=
(AddGroupWithOne.intCast_ofNat 0).trans Nat.cast_zero
#align int.cast_zero Int.cast_zeroₓ
-- type had `HasLiftT`
@[simp high, nolint simpNF, norm_cast] -- this lemma competes with `Int.ofNat_eq_cast` to come later
theorem cast_ofNat (n : ℕ) : ((n : ℤ) : R) = n :=
AddGroupWithOne.intCast_ofNat _
#align int.cast_coe_nat Int.cast_ofNatₓ
-- expected `n` to be implicit, and `HasLiftT`
#align int.cast_of_nat Int.cast_ofNatₓ
-- See note [no_index around OfNat.ofNat]
@[simp, norm_cast]
theorem int_cast_ofNat (n : ℕ) [n.AtLeastTwo] :
((no_index (OfNat.ofNat n) : ℤ) : R) = OfNat.ofNat n := by
simpa only [OfNat.ofNat] using AddGroupWithOne.intCast_ofNat (R := R) n
@[simp, norm_cast]
theorem cast_one : ((1 : ℤ) : R) = 1 := by
erw [cast_ofNat, Nat.cast_one]
#align int.cast_one Int.cast_oneₓ
-- type had `HasLiftT`
@[simp, norm_cast]
theorem cast_neg : ∀ n, ((-n : ℤ) : R) = -n
| (0 : ℕ) => by erw [cast_zero, neg_zero]
| (n + 1 : ℕ) => by erw [cast_ofNat, cast_negSucc]
| -[n+1] => by erw [cast_ofNat, cast_negSucc, neg_neg]
#align int.cast_neg Int.cast_negₓ
-- type had `HasLiftT`
@[simp, norm_cast]
theorem cast_subNatNat (m n) : ((Int.subNatNat m n : ℤ) : R) = m - n := by
unfold subNatNat
|
cases e : n - m
|
@[simp, norm_cast]
theorem cast_subNatNat (m n) : ((Int.subNatNat m n : ℤ) : R) = m - n := by
unfold subNatNat
|
Mathlib.Data.Int.Cast.Basic.91_0.3MsWc9B5PAFbTbn
|
@[simp, norm_cast]
theorem cast_subNatNat (m n) : ((Int.subNatNat m n : ℤ) : R) = m - n
|
Mathlib_Data_Int_Cast_Basic
|
case zero
R : Type u
inst✝ : AddGroupWithOne R
m n : ℕ
e : n - m = zero
⊢ ↑(match zero with
| 0 => ofNat (m - n)
| succ k => -[k+1]) =
↑m - ↑n
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Cast of integers (additional theorems)
This file proves additional properties about the *canonical* homomorphism from
the integers into an additive group with a one (`Int.cast`).
There is also `Data.Int.Cast.Lemmas`,
which includes lemmas stated in terms of algebraic homomorphisms,
and results involving the order structure of `ℤ`.
By contrast, this file's only import beyond `Data.Int.Cast.Defs` is `Algebra.Group.Basic`.
-/
universe u
namespace Nat
variable {R : Type u} [AddGroupWithOne R]
@[simp, norm_cast]
theorem cast_sub {m n} (h : m ≤ n) : ((n - m : ℕ) : R) = n - m :=
eq_sub_of_add_eq <| by rw [← cast_add, Nat.sub_add_cancel h]
#align nat.cast_sub Nat.cast_subₓ
-- `HasLiftT` appeared in the type signature
@[simp, norm_cast]
theorem cast_pred : ∀ {n}, 0 < n → ((n - 1 : ℕ) : R) = n - 1
| 0, h => by cases h
| n + 1, _ => by rw [cast_succ, add_sub_cancel]; rfl
#align nat.cast_pred Nat.cast_pred
end Nat
open Nat
namespace Int
variable {R : Type u} [AddGroupWithOne R]
@[simp, norm_cast squash]
theorem cast_negSucc (n : ℕ) : (-[n+1] : R) = -(n + 1 : ℕ) :=
AddGroupWithOne.intCast_negSucc n
#align int.cast_neg_succ_of_nat Int.cast_negSuccₓ
-- expected `n` to be implicit, and `HasLiftT`
@[simp, norm_cast]
theorem cast_zero : ((0 : ℤ) : R) = 0 :=
(AddGroupWithOne.intCast_ofNat 0).trans Nat.cast_zero
#align int.cast_zero Int.cast_zeroₓ
-- type had `HasLiftT`
@[simp high, nolint simpNF, norm_cast] -- this lemma competes with `Int.ofNat_eq_cast` to come later
theorem cast_ofNat (n : ℕ) : ((n : ℤ) : R) = n :=
AddGroupWithOne.intCast_ofNat _
#align int.cast_coe_nat Int.cast_ofNatₓ
-- expected `n` to be implicit, and `HasLiftT`
#align int.cast_of_nat Int.cast_ofNatₓ
-- See note [no_index around OfNat.ofNat]
@[simp, norm_cast]
theorem int_cast_ofNat (n : ℕ) [n.AtLeastTwo] :
((no_index (OfNat.ofNat n) : ℤ) : R) = OfNat.ofNat n := by
simpa only [OfNat.ofNat] using AddGroupWithOne.intCast_ofNat (R := R) n
@[simp, norm_cast]
theorem cast_one : ((1 : ℤ) : R) = 1 := by
erw [cast_ofNat, Nat.cast_one]
#align int.cast_one Int.cast_oneₓ
-- type had `HasLiftT`
@[simp, norm_cast]
theorem cast_neg : ∀ n, ((-n : ℤ) : R) = -n
| (0 : ℕ) => by erw [cast_zero, neg_zero]
| (n + 1 : ℕ) => by erw [cast_ofNat, cast_negSucc]
| -[n+1] => by erw [cast_ofNat, cast_negSucc, neg_neg]
#align int.cast_neg Int.cast_negₓ
-- type had `HasLiftT`
@[simp, norm_cast]
theorem cast_subNatNat (m n) : ((Int.subNatNat m n : ℤ) : R) = m - n := by
unfold subNatNat
cases e : n - m
·
|
simp only [ofNat_eq_coe]
|
@[simp, norm_cast]
theorem cast_subNatNat (m n) : ((Int.subNatNat m n : ℤ) : R) = m - n := by
unfold subNatNat
cases e : n - m
·
|
Mathlib.Data.Int.Cast.Basic.91_0.3MsWc9B5PAFbTbn
|
@[simp, norm_cast]
theorem cast_subNatNat (m n) : ((Int.subNatNat m n : ℤ) : R) = m - n
|
Mathlib_Data_Int_Cast_Basic
|
case zero
R : Type u
inst✝ : AddGroupWithOne R
m n : ℕ
e : n - m = zero
⊢ ↑↑(m - n) = ↑m - ↑n
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Cast of integers (additional theorems)
This file proves additional properties about the *canonical* homomorphism from
the integers into an additive group with a one (`Int.cast`).
There is also `Data.Int.Cast.Lemmas`,
which includes lemmas stated in terms of algebraic homomorphisms,
and results involving the order structure of `ℤ`.
By contrast, this file's only import beyond `Data.Int.Cast.Defs` is `Algebra.Group.Basic`.
-/
universe u
namespace Nat
variable {R : Type u} [AddGroupWithOne R]
@[simp, norm_cast]
theorem cast_sub {m n} (h : m ≤ n) : ((n - m : ℕ) : R) = n - m :=
eq_sub_of_add_eq <| by rw [← cast_add, Nat.sub_add_cancel h]
#align nat.cast_sub Nat.cast_subₓ
-- `HasLiftT` appeared in the type signature
@[simp, norm_cast]
theorem cast_pred : ∀ {n}, 0 < n → ((n - 1 : ℕ) : R) = n - 1
| 0, h => by cases h
| n + 1, _ => by rw [cast_succ, add_sub_cancel]; rfl
#align nat.cast_pred Nat.cast_pred
end Nat
open Nat
namespace Int
variable {R : Type u} [AddGroupWithOne R]
@[simp, norm_cast squash]
theorem cast_negSucc (n : ℕ) : (-[n+1] : R) = -(n + 1 : ℕ) :=
AddGroupWithOne.intCast_negSucc n
#align int.cast_neg_succ_of_nat Int.cast_negSuccₓ
-- expected `n` to be implicit, and `HasLiftT`
@[simp, norm_cast]
theorem cast_zero : ((0 : ℤ) : R) = 0 :=
(AddGroupWithOne.intCast_ofNat 0).trans Nat.cast_zero
#align int.cast_zero Int.cast_zeroₓ
-- type had `HasLiftT`
@[simp high, nolint simpNF, norm_cast] -- this lemma competes with `Int.ofNat_eq_cast` to come later
theorem cast_ofNat (n : ℕ) : ((n : ℤ) : R) = n :=
AddGroupWithOne.intCast_ofNat _
#align int.cast_coe_nat Int.cast_ofNatₓ
-- expected `n` to be implicit, and `HasLiftT`
#align int.cast_of_nat Int.cast_ofNatₓ
-- See note [no_index around OfNat.ofNat]
@[simp, norm_cast]
theorem int_cast_ofNat (n : ℕ) [n.AtLeastTwo] :
((no_index (OfNat.ofNat n) : ℤ) : R) = OfNat.ofNat n := by
simpa only [OfNat.ofNat] using AddGroupWithOne.intCast_ofNat (R := R) n
@[simp, norm_cast]
theorem cast_one : ((1 : ℤ) : R) = 1 := by
erw [cast_ofNat, Nat.cast_one]
#align int.cast_one Int.cast_oneₓ
-- type had `HasLiftT`
@[simp, norm_cast]
theorem cast_neg : ∀ n, ((-n : ℤ) : R) = -n
| (0 : ℕ) => by erw [cast_zero, neg_zero]
| (n + 1 : ℕ) => by erw [cast_ofNat, cast_negSucc]
| -[n+1] => by erw [cast_ofNat, cast_negSucc, neg_neg]
#align int.cast_neg Int.cast_negₓ
-- type had `HasLiftT`
@[simp, norm_cast]
theorem cast_subNatNat (m n) : ((Int.subNatNat m n : ℤ) : R) = m - n := by
unfold subNatNat
cases e : n - m
· simp only [ofNat_eq_coe]
|
simp [e, Nat.le_of_sub_eq_zero e]
|
@[simp, norm_cast]
theorem cast_subNatNat (m n) : ((Int.subNatNat m n : ℤ) : R) = m - n := by
unfold subNatNat
cases e : n - m
· simp only [ofNat_eq_coe]
|
Mathlib.Data.Int.Cast.Basic.91_0.3MsWc9B5PAFbTbn
|
@[simp, norm_cast]
theorem cast_subNatNat (m n) : ((Int.subNatNat m n : ℤ) : R) = m - n
|
Mathlib_Data_Int_Cast_Basic
|
case succ
R : Type u
inst✝ : AddGroupWithOne R
m n n✝ : ℕ
e : n - m = succ n✝
⊢ ↑(match succ n✝ with
| 0 => ofNat (m - n)
| succ k => -[k+1]) =
↑m - ↑n
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Cast of integers (additional theorems)
This file proves additional properties about the *canonical* homomorphism from
the integers into an additive group with a one (`Int.cast`).
There is also `Data.Int.Cast.Lemmas`,
which includes lemmas stated in terms of algebraic homomorphisms,
and results involving the order structure of `ℤ`.
By contrast, this file's only import beyond `Data.Int.Cast.Defs` is `Algebra.Group.Basic`.
-/
universe u
namespace Nat
variable {R : Type u} [AddGroupWithOne R]
@[simp, norm_cast]
theorem cast_sub {m n} (h : m ≤ n) : ((n - m : ℕ) : R) = n - m :=
eq_sub_of_add_eq <| by rw [← cast_add, Nat.sub_add_cancel h]
#align nat.cast_sub Nat.cast_subₓ
-- `HasLiftT` appeared in the type signature
@[simp, norm_cast]
theorem cast_pred : ∀ {n}, 0 < n → ((n - 1 : ℕ) : R) = n - 1
| 0, h => by cases h
| n + 1, _ => by rw [cast_succ, add_sub_cancel]; rfl
#align nat.cast_pred Nat.cast_pred
end Nat
open Nat
namespace Int
variable {R : Type u} [AddGroupWithOne R]
@[simp, norm_cast squash]
theorem cast_negSucc (n : ℕ) : (-[n+1] : R) = -(n + 1 : ℕ) :=
AddGroupWithOne.intCast_negSucc n
#align int.cast_neg_succ_of_nat Int.cast_negSuccₓ
-- expected `n` to be implicit, and `HasLiftT`
@[simp, norm_cast]
theorem cast_zero : ((0 : ℤ) : R) = 0 :=
(AddGroupWithOne.intCast_ofNat 0).trans Nat.cast_zero
#align int.cast_zero Int.cast_zeroₓ
-- type had `HasLiftT`
@[simp high, nolint simpNF, norm_cast] -- this lemma competes with `Int.ofNat_eq_cast` to come later
theorem cast_ofNat (n : ℕ) : ((n : ℤ) : R) = n :=
AddGroupWithOne.intCast_ofNat _
#align int.cast_coe_nat Int.cast_ofNatₓ
-- expected `n` to be implicit, and `HasLiftT`
#align int.cast_of_nat Int.cast_ofNatₓ
-- See note [no_index around OfNat.ofNat]
@[simp, norm_cast]
theorem int_cast_ofNat (n : ℕ) [n.AtLeastTwo] :
((no_index (OfNat.ofNat n) : ℤ) : R) = OfNat.ofNat n := by
simpa only [OfNat.ofNat] using AddGroupWithOne.intCast_ofNat (R := R) n
@[simp, norm_cast]
theorem cast_one : ((1 : ℤ) : R) = 1 := by
erw [cast_ofNat, Nat.cast_one]
#align int.cast_one Int.cast_oneₓ
-- type had `HasLiftT`
@[simp, norm_cast]
theorem cast_neg : ∀ n, ((-n : ℤ) : R) = -n
| (0 : ℕ) => by erw [cast_zero, neg_zero]
| (n + 1 : ℕ) => by erw [cast_ofNat, cast_negSucc]
| -[n+1] => by erw [cast_ofNat, cast_negSucc, neg_neg]
#align int.cast_neg Int.cast_negₓ
-- type had `HasLiftT`
@[simp, norm_cast]
theorem cast_subNatNat (m n) : ((Int.subNatNat m n : ℤ) : R) = m - n := by
unfold subNatNat
cases e : n - m
· simp only [ofNat_eq_coe]
simp [e, Nat.le_of_sub_eq_zero e]
·
|
rw [cast_negSucc, Nat.add_one, ← e, Nat.cast_sub <| _root_.le_of_lt <| Nat.lt_of_sub_eq_succ e,
neg_sub]
|
@[simp, norm_cast]
theorem cast_subNatNat (m n) : ((Int.subNatNat m n : ℤ) : R) = m - n := by
unfold subNatNat
cases e : n - m
· simp only [ofNat_eq_coe]
simp [e, Nat.le_of_sub_eq_zero e]
·
|
Mathlib.Data.Int.Cast.Basic.91_0.3MsWc9B5PAFbTbn
|
@[simp, norm_cast]
theorem cast_subNatNat (m n) : ((Int.subNatNat m n : ℤ) : R) = m - n
|
Mathlib_Data_Int_Cast_Basic
|
R : Type u
inst✝ : AddGroupWithOne R
n : ℕ
⊢ ↑(negOfNat n) = -↑n
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Cast of integers (additional theorems)
This file proves additional properties about the *canonical* homomorphism from
the integers into an additive group with a one (`Int.cast`).
There is also `Data.Int.Cast.Lemmas`,
which includes lemmas stated in terms of algebraic homomorphisms,
and results involving the order structure of `ℤ`.
By contrast, this file's only import beyond `Data.Int.Cast.Defs` is `Algebra.Group.Basic`.
-/
universe u
namespace Nat
variable {R : Type u} [AddGroupWithOne R]
@[simp, norm_cast]
theorem cast_sub {m n} (h : m ≤ n) : ((n - m : ℕ) : R) = n - m :=
eq_sub_of_add_eq <| by rw [← cast_add, Nat.sub_add_cancel h]
#align nat.cast_sub Nat.cast_subₓ
-- `HasLiftT` appeared in the type signature
@[simp, norm_cast]
theorem cast_pred : ∀ {n}, 0 < n → ((n - 1 : ℕ) : R) = n - 1
| 0, h => by cases h
| n + 1, _ => by rw [cast_succ, add_sub_cancel]; rfl
#align nat.cast_pred Nat.cast_pred
end Nat
open Nat
namespace Int
variable {R : Type u} [AddGroupWithOne R]
@[simp, norm_cast squash]
theorem cast_negSucc (n : ℕ) : (-[n+1] : R) = -(n + 1 : ℕ) :=
AddGroupWithOne.intCast_negSucc n
#align int.cast_neg_succ_of_nat Int.cast_negSuccₓ
-- expected `n` to be implicit, and `HasLiftT`
@[simp, norm_cast]
theorem cast_zero : ((0 : ℤ) : R) = 0 :=
(AddGroupWithOne.intCast_ofNat 0).trans Nat.cast_zero
#align int.cast_zero Int.cast_zeroₓ
-- type had `HasLiftT`
@[simp high, nolint simpNF, norm_cast] -- this lemma competes with `Int.ofNat_eq_cast` to come later
theorem cast_ofNat (n : ℕ) : ((n : ℤ) : R) = n :=
AddGroupWithOne.intCast_ofNat _
#align int.cast_coe_nat Int.cast_ofNatₓ
-- expected `n` to be implicit, and `HasLiftT`
#align int.cast_of_nat Int.cast_ofNatₓ
-- See note [no_index around OfNat.ofNat]
@[simp, norm_cast]
theorem int_cast_ofNat (n : ℕ) [n.AtLeastTwo] :
((no_index (OfNat.ofNat n) : ℤ) : R) = OfNat.ofNat n := by
simpa only [OfNat.ofNat] using AddGroupWithOne.intCast_ofNat (R := R) n
@[simp, norm_cast]
theorem cast_one : ((1 : ℤ) : R) = 1 := by
erw [cast_ofNat, Nat.cast_one]
#align int.cast_one Int.cast_oneₓ
-- type had `HasLiftT`
@[simp, norm_cast]
theorem cast_neg : ∀ n, ((-n : ℤ) : R) = -n
| (0 : ℕ) => by erw [cast_zero, neg_zero]
| (n + 1 : ℕ) => by erw [cast_ofNat, cast_negSucc]
| -[n+1] => by erw [cast_ofNat, cast_negSucc, neg_neg]
#align int.cast_neg Int.cast_negₓ
-- type had `HasLiftT`
@[simp, norm_cast]
theorem cast_subNatNat (m n) : ((Int.subNatNat m n : ℤ) : R) = m - n := by
unfold subNatNat
cases e : n - m
· simp only [ofNat_eq_coe]
simp [e, Nat.le_of_sub_eq_zero e]
· rw [cast_negSucc, Nat.add_one, ← e, Nat.cast_sub <| _root_.le_of_lt <| Nat.lt_of_sub_eq_succ e,
neg_sub]
#align int.cast_sub_nat_nat Int.cast_subNatNatₓ
-- type had `HasLiftT`
#align int.neg_of_nat_eq Int.negOfNat_eq
@[simp]
theorem cast_negOfNat (n : ℕ) : ((negOfNat n : ℤ) : R) = -n := by
|
simp [Int.cast_neg, negOfNat_eq]
|
@[simp]
theorem cast_negOfNat (n : ℕ) : ((negOfNat n : ℤ) : R) = -n := by
|
Mathlib.Data.Int.Cast.Basic.104_0.3MsWc9B5PAFbTbn
|
@[simp]
theorem cast_negOfNat (n : ℕ) : ((negOfNat n : ℤ) : R) = -n
|
Mathlib_Data_Int_Cast_Basic
|
R : Type u
inst✝ : AddGroupWithOne R
m n : ℕ
⊢ ↑(↑m + ↑n) = ↑↑m + ↑↑n
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Cast of integers (additional theorems)
This file proves additional properties about the *canonical* homomorphism from
the integers into an additive group with a one (`Int.cast`).
There is also `Data.Int.Cast.Lemmas`,
which includes lemmas stated in terms of algebraic homomorphisms,
and results involving the order structure of `ℤ`.
By contrast, this file's only import beyond `Data.Int.Cast.Defs` is `Algebra.Group.Basic`.
-/
universe u
namespace Nat
variable {R : Type u} [AddGroupWithOne R]
@[simp, norm_cast]
theorem cast_sub {m n} (h : m ≤ n) : ((n - m : ℕ) : R) = n - m :=
eq_sub_of_add_eq <| by rw [← cast_add, Nat.sub_add_cancel h]
#align nat.cast_sub Nat.cast_subₓ
-- `HasLiftT` appeared in the type signature
@[simp, norm_cast]
theorem cast_pred : ∀ {n}, 0 < n → ((n - 1 : ℕ) : R) = n - 1
| 0, h => by cases h
| n + 1, _ => by rw [cast_succ, add_sub_cancel]; rfl
#align nat.cast_pred Nat.cast_pred
end Nat
open Nat
namespace Int
variable {R : Type u} [AddGroupWithOne R]
@[simp, norm_cast squash]
theorem cast_negSucc (n : ℕ) : (-[n+1] : R) = -(n + 1 : ℕ) :=
AddGroupWithOne.intCast_negSucc n
#align int.cast_neg_succ_of_nat Int.cast_negSuccₓ
-- expected `n` to be implicit, and `HasLiftT`
@[simp, norm_cast]
theorem cast_zero : ((0 : ℤ) : R) = 0 :=
(AddGroupWithOne.intCast_ofNat 0).trans Nat.cast_zero
#align int.cast_zero Int.cast_zeroₓ
-- type had `HasLiftT`
@[simp high, nolint simpNF, norm_cast] -- this lemma competes with `Int.ofNat_eq_cast` to come later
theorem cast_ofNat (n : ℕ) : ((n : ℤ) : R) = n :=
AddGroupWithOne.intCast_ofNat _
#align int.cast_coe_nat Int.cast_ofNatₓ
-- expected `n` to be implicit, and `HasLiftT`
#align int.cast_of_nat Int.cast_ofNatₓ
-- See note [no_index around OfNat.ofNat]
@[simp, norm_cast]
theorem int_cast_ofNat (n : ℕ) [n.AtLeastTwo] :
((no_index (OfNat.ofNat n) : ℤ) : R) = OfNat.ofNat n := by
simpa only [OfNat.ofNat] using AddGroupWithOne.intCast_ofNat (R := R) n
@[simp, norm_cast]
theorem cast_one : ((1 : ℤ) : R) = 1 := by
erw [cast_ofNat, Nat.cast_one]
#align int.cast_one Int.cast_oneₓ
-- type had `HasLiftT`
@[simp, norm_cast]
theorem cast_neg : ∀ n, ((-n : ℤ) : R) = -n
| (0 : ℕ) => by erw [cast_zero, neg_zero]
| (n + 1 : ℕ) => by erw [cast_ofNat, cast_negSucc]
| -[n+1] => by erw [cast_ofNat, cast_negSucc, neg_neg]
#align int.cast_neg Int.cast_negₓ
-- type had `HasLiftT`
@[simp, norm_cast]
theorem cast_subNatNat (m n) : ((Int.subNatNat m n : ℤ) : R) = m - n := by
unfold subNatNat
cases e : n - m
· simp only [ofNat_eq_coe]
simp [e, Nat.le_of_sub_eq_zero e]
· rw [cast_negSucc, Nat.add_one, ← e, Nat.cast_sub <| _root_.le_of_lt <| Nat.lt_of_sub_eq_succ e,
neg_sub]
#align int.cast_sub_nat_nat Int.cast_subNatNatₓ
-- type had `HasLiftT`
#align int.neg_of_nat_eq Int.negOfNat_eq
@[simp]
theorem cast_negOfNat (n : ℕ) : ((negOfNat n : ℤ) : R) = -n := by simp [Int.cast_neg, negOfNat_eq]
#align int.cast_neg_of_nat Int.cast_negOfNat
@[simp, norm_cast]
theorem cast_add : ∀ m n, ((m + n : ℤ) : R) = m + n
| (m : ℕ), (n : ℕ) => by
|
simp [-Int.natCast_add, ← Int.ofNat_add]
|
@[simp, norm_cast]
theorem cast_add : ∀ m n, ((m + n : ℤ) : R) = m + n
| (m : ℕ), (n : ℕ) => by
|
Mathlib.Data.Int.Cast.Basic.108_0.3MsWc9B5PAFbTbn
|
@[simp, norm_cast]
theorem cast_add : ∀ m n, ((m + n : ℤ) : R) = m + n
| (m : ℕ), (n : ℕ) => by simp [-Int.natCast_add, ← Int.ofNat_add]
| (m : ℕ), -[n+1] => by erw [cast_subNatNat, cast_ofNat, cast_negSucc, sub_eq_add_neg]
| -[m+1], (n : ℕ) => by
erw [cast_subNatNat, cast_ofNat, cast_negSucc, sub_eq_iff_eq_add, add_assoc,
eq_neg_add_iff_add_eq, ← Nat.cast_add, ← Nat.cast_add, Nat.add_comm]
| -[m+1], -[n+1] =>
show (-[m + n + 1+1] : R) = _ by
rw [cast_negSucc, cast_negSucc, cast_negSucc, ← neg_add_rev, ← Nat.cast_add,
Nat.add_right_comm m n 1, Nat.add_assoc, Nat.add_comm]
|
Mathlib_Data_Int_Cast_Basic
|
R : Type u
inst✝ : AddGroupWithOne R
m n : ℕ
⊢ ↑(↑m + -[n+1]) = ↑↑m + ↑-[n+1]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Cast of integers (additional theorems)
This file proves additional properties about the *canonical* homomorphism from
the integers into an additive group with a one (`Int.cast`).
There is also `Data.Int.Cast.Lemmas`,
which includes lemmas stated in terms of algebraic homomorphisms,
and results involving the order structure of `ℤ`.
By contrast, this file's only import beyond `Data.Int.Cast.Defs` is `Algebra.Group.Basic`.
-/
universe u
namespace Nat
variable {R : Type u} [AddGroupWithOne R]
@[simp, norm_cast]
theorem cast_sub {m n} (h : m ≤ n) : ((n - m : ℕ) : R) = n - m :=
eq_sub_of_add_eq <| by rw [← cast_add, Nat.sub_add_cancel h]
#align nat.cast_sub Nat.cast_subₓ
-- `HasLiftT` appeared in the type signature
@[simp, norm_cast]
theorem cast_pred : ∀ {n}, 0 < n → ((n - 1 : ℕ) : R) = n - 1
| 0, h => by cases h
| n + 1, _ => by rw [cast_succ, add_sub_cancel]; rfl
#align nat.cast_pred Nat.cast_pred
end Nat
open Nat
namespace Int
variable {R : Type u} [AddGroupWithOne R]
@[simp, norm_cast squash]
theorem cast_negSucc (n : ℕ) : (-[n+1] : R) = -(n + 1 : ℕ) :=
AddGroupWithOne.intCast_negSucc n
#align int.cast_neg_succ_of_nat Int.cast_negSuccₓ
-- expected `n` to be implicit, and `HasLiftT`
@[simp, norm_cast]
theorem cast_zero : ((0 : ℤ) : R) = 0 :=
(AddGroupWithOne.intCast_ofNat 0).trans Nat.cast_zero
#align int.cast_zero Int.cast_zeroₓ
-- type had `HasLiftT`
@[simp high, nolint simpNF, norm_cast] -- this lemma competes with `Int.ofNat_eq_cast` to come later
theorem cast_ofNat (n : ℕ) : ((n : ℤ) : R) = n :=
AddGroupWithOne.intCast_ofNat _
#align int.cast_coe_nat Int.cast_ofNatₓ
-- expected `n` to be implicit, and `HasLiftT`
#align int.cast_of_nat Int.cast_ofNatₓ
-- See note [no_index around OfNat.ofNat]
@[simp, norm_cast]
theorem int_cast_ofNat (n : ℕ) [n.AtLeastTwo] :
((no_index (OfNat.ofNat n) : ℤ) : R) = OfNat.ofNat n := by
simpa only [OfNat.ofNat] using AddGroupWithOne.intCast_ofNat (R := R) n
@[simp, norm_cast]
theorem cast_one : ((1 : ℤ) : R) = 1 := by
erw [cast_ofNat, Nat.cast_one]
#align int.cast_one Int.cast_oneₓ
-- type had `HasLiftT`
@[simp, norm_cast]
theorem cast_neg : ∀ n, ((-n : ℤ) : R) = -n
| (0 : ℕ) => by erw [cast_zero, neg_zero]
| (n + 1 : ℕ) => by erw [cast_ofNat, cast_negSucc]
| -[n+1] => by erw [cast_ofNat, cast_negSucc, neg_neg]
#align int.cast_neg Int.cast_negₓ
-- type had `HasLiftT`
@[simp, norm_cast]
theorem cast_subNatNat (m n) : ((Int.subNatNat m n : ℤ) : R) = m - n := by
unfold subNatNat
cases e : n - m
· simp only [ofNat_eq_coe]
simp [e, Nat.le_of_sub_eq_zero e]
· rw [cast_negSucc, Nat.add_one, ← e, Nat.cast_sub <| _root_.le_of_lt <| Nat.lt_of_sub_eq_succ e,
neg_sub]
#align int.cast_sub_nat_nat Int.cast_subNatNatₓ
-- type had `HasLiftT`
#align int.neg_of_nat_eq Int.negOfNat_eq
@[simp]
theorem cast_negOfNat (n : ℕ) : ((negOfNat n : ℤ) : R) = -n := by simp [Int.cast_neg, negOfNat_eq]
#align int.cast_neg_of_nat Int.cast_negOfNat
@[simp, norm_cast]
theorem cast_add : ∀ m n, ((m + n : ℤ) : R) = m + n
| (m : ℕ), (n : ℕ) => by simp [-Int.natCast_add, ← Int.ofNat_add]
| (m : ℕ), -[n+1] => by
|
erw [cast_subNatNat, cast_ofNat, cast_negSucc, sub_eq_add_neg]
|
@[simp, norm_cast]
theorem cast_add : ∀ m n, ((m + n : ℤ) : R) = m + n
| (m : ℕ), (n : ℕ) => by simp [-Int.natCast_add, ← Int.ofNat_add]
| (m : ℕ), -[n+1] => by
|
Mathlib.Data.Int.Cast.Basic.108_0.3MsWc9B5PAFbTbn
|
@[simp, norm_cast]
theorem cast_add : ∀ m n, ((m + n : ℤ) : R) = m + n
| (m : ℕ), (n : ℕ) => by simp [-Int.natCast_add, ← Int.ofNat_add]
| (m : ℕ), -[n+1] => by erw [cast_subNatNat, cast_ofNat, cast_negSucc, sub_eq_add_neg]
| -[m+1], (n : ℕ) => by
erw [cast_subNatNat, cast_ofNat, cast_negSucc, sub_eq_iff_eq_add, add_assoc,
eq_neg_add_iff_add_eq, ← Nat.cast_add, ← Nat.cast_add, Nat.add_comm]
| -[m+1], -[n+1] =>
show (-[m + n + 1+1] : R) = _ by
rw [cast_negSucc, cast_negSucc, cast_negSucc, ← neg_add_rev, ← Nat.cast_add,
Nat.add_right_comm m n 1, Nat.add_assoc, Nat.add_comm]
|
Mathlib_Data_Int_Cast_Basic
|
R : Type u
inst✝ : AddGroupWithOne R
m n : ℕ
⊢ ↑(-[m+1] + ↑n) = ↑-[m+1] + ↑↑n
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Cast of integers (additional theorems)
This file proves additional properties about the *canonical* homomorphism from
the integers into an additive group with a one (`Int.cast`).
There is also `Data.Int.Cast.Lemmas`,
which includes lemmas stated in terms of algebraic homomorphisms,
and results involving the order structure of `ℤ`.
By contrast, this file's only import beyond `Data.Int.Cast.Defs` is `Algebra.Group.Basic`.
-/
universe u
namespace Nat
variable {R : Type u} [AddGroupWithOne R]
@[simp, norm_cast]
theorem cast_sub {m n} (h : m ≤ n) : ((n - m : ℕ) : R) = n - m :=
eq_sub_of_add_eq <| by rw [← cast_add, Nat.sub_add_cancel h]
#align nat.cast_sub Nat.cast_subₓ
-- `HasLiftT` appeared in the type signature
@[simp, norm_cast]
theorem cast_pred : ∀ {n}, 0 < n → ((n - 1 : ℕ) : R) = n - 1
| 0, h => by cases h
| n + 1, _ => by rw [cast_succ, add_sub_cancel]; rfl
#align nat.cast_pred Nat.cast_pred
end Nat
open Nat
namespace Int
variable {R : Type u} [AddGroupWithOne R]
@[simp, norm_cast squash]
theorem cast_negSucc (n : ℕ) : (-[n+1] : R) = -(n + 1 : ℕ) :=
AddGroupWithOne.intCast_negSucc n
#align int.cast_neg_succ_of_nat Int.cast_negSuccₓ
-- expected `n` to be implicit, and `HasLiftT`
@[simp, norm_cast]
theorem cast_zero : ((0 : ℤ) : R) = 0 :=
(AddGroupWithOne.intCast_ofNat 0).trans Nat.cast_zero
#align int.cast_zero Int.cast_zeroₓ
-- type had `HasLiftT`
@[simp high, nolint simpNF, norm_cast] -- this lemma competes with `Int.ofNat_eq_cast` to come later
theorem cast_ofNat (n : ℕ) : ((n : ℤ) : R) = n :=
AddGroupWithOne.intCast_ofNat _
#align int.cast_coe_nat Int.cast_ofNatₓ
-- expected `n` to be implicit, and `HasLiftT`
#align int.cast_of_nat Int.cast_ofNatₓ
-- See note [no_index around OfNat.ofNat]
@[simp, norm_cast]
theorem int_cast_ofNat (n : ℕ) [n.AtLeastTwo] :
((no_index (OfNat.ofNat n) : ℤ) : R) = OfNat.ofNat n := by
simpa only [OfNat.ofNat] using AddGroupWithOne.intCast_ofNat (R := R) n
@[simp, norm_cast]
theorem cast_one : ((1 : ℤ) : R) = 1 := by
erw [cast_ofNat, Nat.cast_one]
#align int.cast_one Int.cast_oneₓ
-- type had `HasLiftT`
@[simp, norm_cast]
theorem cast_neg : ∀ n, ((-n : ℤ) : R) = -n
| (0 : ℕ) => by erw [cast_zero, neg_zero]
| (n + 1 : ℕ) => by erw [cast_ofNat, cast_negSucc]
| -[n+1] => by erw [cast_ofNat, cast_negSucc, neg_neg]
#align int.cast_neg Int.cast_negₓ
-- type had `HasLiftT`
@[simp, norm_cast]
theorem cast_subNatNat (m n) : ((Int.subNatNat m n : ℤ) : R) = m - n := by
unfold subNatNat
cases e : n - m
· simp only [ofNat_eq_coe]
simp [e, Nat.le_of_sub_eq_zero e]
· rw [cast_negSucc, Nat.add_one, ← e, Nat.cast_sub <| _root_.le_of_lt <| Nat.lt_of_sub_eq_succ e,
neg_sub]
#align int.cast_sub_nat_nat Int.cast_subNatNatₓ
-- type had `HasLiftT`
#align int.neg_of_nat_eq Int.negOfNat_eq
@[simp]
theorem cast_negOfNat (n : ℕ) : ((negOfNat n : ℤ) : R) = -n := by simp [Int.cast_neg, negOfNat_eq]
#align int.cast_neg_of_nat Int.cast_negOfNat
@[simp, norm_cast]
theorem cast_add : ∀ m n, ((m + n : ℤ) : R) = m + n
| (m : ℕ), (n : ℕ) => by simp [-Int.natCast_add, ← Int.ofNat_add]
| (m : ℕ), -[n+1] => by erw [cast_subNatNat, cast_ofNat, cast_negSucc, sub_eq_add_neg]
| -[m+1], (n : ℕ) => by
|
erw [cast_subNatNat, cast_ofNat, cast_negSucc, sub_eq_iff_eq_add, add_assoc,
eq_neg_add_iff_add_eq, ← Nat.cast_add, ← Nat.cast_add, Nat.add_comm]
|
@[simp, norm_cast]
theorem cast_add : ∀ m n, ((m + n : ℤ) : R) = m + n
| (m : ℕ), (n : ℕ) => by simp [-Int.natCast_add, ← Int.ofNat_add]
| (m : ℕ), -[n+1] => by erw [cast_subNatNat, cast_ofNat, cast_negSucc, sub_eq_add_neg]
| -[m+1], (n : ℕ) => by
|
Mathlib.Data.Int.Cast.Basic.108_0.3MsWc9B5PAFbTbn
|
@[simp, norm_cast]
theorem cast_add : ∀ m n, ((m + n : ℤ) : R) = m + n
| (m : ℕ), (n : ℕ) => by simp [-Int.natCast_add, ← Int.ofNat_add]
| (m : ℕ), -[n+1] => by erw [cast_subNatNat, cast_ofNat, cast_negSucc, sub_eq_add_neg]
| -[m+1], (n : ℕ) => by
erw [cast_subNatNat, cast_ofNat, cast_negSucc, sub_eq_iff_eq_add, add_assoc,
eq_neg_add_iff_add_eq, ← Nat.cast_add, ← Nat.cast_add, Nat.add_comm]
| -[m+1], -[n+1] =>
show (-[m + n + 1+1] : R) = _ by
rw [cast_negSucc, cast_negSucc, cast_negSucc, ← neg_add_rev, ← Nat.cast_add,
Nat.add_right_comm m n 1, Nat.add_assoc, Nat.add_comm]
|
Mathlib_Data_Int_Cast_Basic
|
R : Type u
inst✝ : AddGroupWithOne R
m n : ℕ
⊢ ↑-[m + n + 1+1] = ↑-[m+1] + ↑-[n+1]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Cast of integers (additional theorems)
This file proves additional properties about the *canonical* homomorphism from
the integers into an additive group with a one (`Int.cast`).
There is also `Data.Int.Cast.Lemmas`,
which includes lemmas stated in terms of algebraic homomorphisms,
and results involving the order structure of `ℤ`.
By contrast, this file's only import beyond `Data.Int.Cast.Defs` is `Algebra.Group.Basic`.
-/
universe u
namespace Nat
variable {R : Type u} [AddGroupWithOne R]
@[simp, norm_cast]
theorem cast_sub {m n} (h : m ≤ n) : ((n - m : ℕ) : R) = n - m :=
eq_sub_of_add_eq <| by rw [← cast_add, Nat.sub_add_cancel h]
#align nat.cast_sub Nat.cast_subₓ
-- `HasLiftT` appeared in the type signature
@[simp, norm_cast]
theorem cast_pred : ∀ {n}, 0 < n → ((n - 1 : ℕ) : R) = n - 1
| 0, h => by cases h
| n + 1, _ => by rw [cast_succ, add_sub_cancel]; rfl
#align nat.cast_pred Nat.cast_pred
end Nat
open Nat
namespace Int
variable {R : Type u} [AddGroupWithOne R]
@[simp, norm_cast squash]
theorem cast_negSucc (n : ℕ) : (-[n+1] : R) = -(n + 1 : ℕ) :=
AddGroupWithOne.intCast_negSucc n
#align int.cast_neg_succ_of_nat Int.cast_negSuccₓ
-- expected `n` to be implicit, and `HasLiftT`
@[simp, norm_cast]
theorem cast_zero : ((0 : ℤ) : R) = 0 :=
(AddGroupWithOne.intCast_ofNat 0).trans Nat.cast_zero
#align int.cast_zero Int.cast_zeroₓ
-- type had `HasLiftT`
@[simp high, nolint simpNF, norm_cast] -- this lemma competes with `Int.ofNat_eq_cast` to come later
theorem cast_ofNat (n : ℕ) : ((n : ℤ) : R) = n :=
AddGroupWithOne.intCast_ofNat _
#align int.cast_coe_nat Int.cast_ofNatₓ
-- expected `n` to be implicit, and `HasLiftT`
#align int.cast_of_nat Int.cast_ofNatₓ
-- See note [no_index around OfNat.ofNat]
@[simp, norm_cast]
theorem int_cast_ofNat (n : ℕ) [n.AtLeastTwo] :
((no_index (OfNat.ofNat n) : ℤ) : R) = OfNat.ofNat n := by
simpa only [OfNat.ofNat] using AddGroupWithOne.intCast_ofNat (R := R) n
@[simp, norm_cast]
theorem cast_one : ((1 : ℤ) : R) = 1 := by
erw [cast_ofNat, Nat.cast_one]
#align int.cast_one Int.cast_oneₓ
-- type had `HasLiftT`
@[simp, norm_cast]
theorem cast_neg : ∀ n, ((-n : ℤ) : R) = -n
| (0 : ℕ) => by erw [cast_zero, neg_zero]
| (n + 1 : ℕ) => by erw [cast_ofNat, cast_negSucc]
| -[n+1] => by erw [cast_ofNat, cast_negSucc, neg_neg]
#align int.cast_neg Int.cast_negₓ
-- type had `HasLiftT`
@[simp, norm_cast]
theorem cast_subNatNat (m n) : ((Int.subNatNat m n : ℤ) : R) = m - n := by
unfold subNatNat
cases e : n - m
· simp only [ofNat_eq_coe]
simp [e, Nat.le_of_sub_eq_zero e]
· rw [cast_negSucc, Nat.add_one, ← e, Nat.cast_sub <| _root_.le_of_lt <| Nat.lt_of_sub_eq_succ e,
neg_sub]
#align int.cast_sub_nat_nat Int.cast_subNatNatₓ
-- type had `HasLiftT`
#align int.neg_of_nat_eq Int.negOfNat_eq
@[simp]
theorem cast_negOfNat (n : ℕ) : ((negOfNat n : ℤ) : R) = -n := by simp [Int.cast_neg, negOfNat_eq]
#align int.cast_neg_of_nat Int.cast_negOfNat
@[simp, norm_cast]
theorem cast_add : ∀ m n, ((m + n : ℤ) : R) = m + n
| (m : ℕ), (n : ℕ) => by simp [-Int.natCast_add, ← Int.ofNat_add]
| (m : ℕ), -[n+1] => by erw [cast_subNatNat, cast_ofNat, cast_negSucc, sub_eq_add_neg]
| -[m+1], (n : ℕ) => by
erw [cast_subNatNat, cast_ofNat, cast_negSucc, sub_eq_iff_eq_add, add_assoc,
eq_neg_add_iff_add_eq, ← Nat.cast_add, ← Nat.cast_add, Nat.add_comm]
| -[m+1], -[n+1] =>
show (-[m + n + 1+1] : R) = _ by
|
rw [cast_negSucc, cast_negSucc, cast_negSucc, ← neg_add_rev, ← Nat.cast_add,
Nat.add_right_comm m n 1, Nat.add_assoc, Nat.add_comm]
|
@[simp, norm_cast]
theorem cast_add : ∀ m n, ((m + n : ℤ) : R) = m + n
| (m : ℕ), (n : ℕ) => by simp [-Int.natCast_add, ← Int.ofNat_add]
| (m : ℕ), -[n+1] => by erw [cast_subNatNat, cast_ofNat, cast_negSucc, sub_eq_add_neg]
| -[m+1], (n : ℕ) => by
erw [cast_subNatNat, cast_ofNat, cast_negSucc, sub_eq_iff_eq_add, add_assoc,
eq_neg_add_iff_add_eq, ← Nat.cast_add, ← Nat.cast_add, Nat.add_comm]
| -[m+1], -[n+1] =>
show (-[m + n + 1+1] : R) = _ by
|
Mathlib.Data.Int.Cast.Basic.108_0.3MsWc9B5PAFbTbn
|
@[simp, norm_cast]
theorem cast_add : ∀ m n, ((m + n : ℤ) : R) = m + n
| (m : ℕ), (n : ℕ) => by simp [-Int.natCast_add, ← Int.ofNat_add]
| (m : ℕ), -[n+1] => by erw [cast_subNatNat, cast_ofNat, cast_negSucc, sub_eq_add_neg]
| -[m+1], (n : ℕ) => by
erw [cast_subNatNat, cast_ofNat, cast_negSucc, sub_eq_iff_eq_add, add_assoc,
eq_neg_add_iff_add_eq, ← Nat.cast_add, ← Nat.cast_add, Nat.add_comm]
| -[m+1], -[n+1] =>
show (-[m + n + 1+1] : R) = _ by
rw [cast_negSucc, cast_negSucc, cast_negSucc, ← neg_add_rev, ← Nat.cast_add,
Nat.add_right_comm m n 1, Nat.add_assoc, Nat.add_comm]
|
Mathlib_Data_Int_Cast_Basic
|
R : Type u
inst✝ : AddGroupWithOne R
m n : ℤ
⊢ ↑(m - n) = ↑m - ↑n
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Cast of integers (additional theorems)
This file proves additional properties about the *canonical* homomorphism from
the integers into an additive group with a one (`Int.cast`).
There is also `Data.Int.Cast.Lemmas`,
which includes lemmas stated in terms of algebraic homomorphisms,
and results involving the order structure of `ℤ`.
By contrast, this file's only import beyond `Data.Int.Cast.Defs` is `Algebra.Group.Basic`.
-/
universe u
namespace Nat
variable {R : Type u} [AddGroupWithOne R]
@[simp, norm_cast]
theorem cast_sub {m n} (h : m ≤ n) : ((n - m : ℕ) : R) = n - m :=
eq_sub_of_add_eq <| by rw [← cast_add, Nat.sub_add_cancel h]
#align nat.cast_sub Nat.cast_subₓ
-- `HasLiftT` appeared in the type signature
@[simp, norm_cast]
theorem cast_pred : ∀ {n}, 0 < n → ((n - 1 : ℕ) : R) = n - 1
| 0, h => by cases h
| n + 1, _ => by rw [cast_succ, add_sub_cancel]; rfl
#align nat.cast_pred Nat.cast_pred
end Nat
open Nat
namespace Int
variable {R : Type u} [AddGroupWithOne R]
@[simp, norm_cast squash]
theorem cast_negSucc (n : ℕ) : (-[n+1] : R) = -(n + 1 : ℕ) :=
AddGroupWithOne.intCast_negSucc n
#align int.cast_neg_succ_of_nat Int.cast_negSuccₓ
-- expected `n` to be implicit, and `HasLiftT`
@[simp, norm_cast]
theorem cast_zero : ((0 : ℤ) : R) = 0 :=
(AddGroupWithOne.intCast_ofNat 0).trans Nat.cast_zero
#align int.cast_zero Int.cast_zeroₓ
-- type had `HasLiftT`
@[simp high, nolint simpNF, norm_cast] -- this lemma competes with `Int.ofNat_eq_cast` to come later
theorem cast_ofNat (n : ℕ) : ((n : ℤ) : R) = n :=
AddGroupWithOne.intCast_ofNat _
#align int.cast_coe_nat Int.cast_ofNatₓ
-- expected `n` to be implicit, and `HasLiftT`
#align int.cast_of_nat Int.cast_ofNatₓ
-- See note [no_index around OfNat.ofNat]
@[simp, norm_cast]
theorem int_cast_ofNat (n : ℕ) [n.AtLeastTwo] :
((no_index (OfNat.ofNat n) : ℤ) : R) = OfNat.ofNat n := by
simpa only [OfNat.ofNat] using AddGroupWithOne.intCast_ofNat (R := R) n
@[simp, norm_cast]
theorem cast_one : ((1 : ℤ) : R) = 1 := by
erw [cast_ofNat, Nat.cast_one]
#align int.cast_one Int.cast_oneₓ
-- type had `HasLiftT`
@[simp, norm_cast]
theorem cast_neg : ∀ n, ((-n : ℤ) : R) = -n
| (0 : ℕ) => by erw [cast_zero, neg_zero]
| (n + 1 : ℕ) => by erw [cast_ofNat, cast_negSucc]
| -[n+1] => by erw [cast_ofNat, cast_negSucc, neg_neg]
#align int.cast_neg Int.cast_negₓ
-- type had `HasLiftT`
@[simp, norm_cast]
theorem cast_subNatNat (m n) : ((Int.subNatNat m n : ℤ) : R) = m - n := by
unfold subNatNat
cases e : n - m
· simp only [ofNat_eq_coe]
simp [e, Nat.le_of_sub_eq_zero e]
· rw [cast_negSucc, Nat.add_one, ← e, Nat.cast_sub <| _root_.le_of_lt <| Nat.lt_of_sub_eq_succ e,
neg_sub]
#align int.cast_sub_nat_nat Int.cast_subNatNatₓ
-- type had `HasLiftT`
#align int.neg_of_nat_eq Int.negOfNat_eq
@[simp]
theorem cast_negOfNat (n : ℕ) : ((negOfNat n : ℤ) : R) = -n := by simp [Int.cast_neg, negOfNat_eq]
#align int.cast_neg_of_nat Int.cast_negOfNat
@[simp, norm_cast]
theorem cast_add : ∀ m n, ((m + n : ℤ) : R) = m + n
| (m : ℕ), (n : ℕ) => by simp [-Int.natCast_add, ← Int.ofNat_add]
| (m : ℕ), -[n+1] => by erw [cast_subNatNat, cast_ofNat, cast_negSucc, sub_eq_add_neg]
| -[m+1], (n : ℕ) => by
erw [cast_subNatNat, cast_ofNat, cast_negSucc, sub_eq_iff_eq_add, add_assoc,
eq_neg_add_iff_add_eq, ← Nat.cast_add, ← Nat.cast_add, Nat.add_comm]
| -[m+1], -[n+1] =>
show (-[m + n + 1+1] : R) = _ by
rw [cast_negSucc, cast_negSucc, cast_negSucc, ← neg_add_rev, ← Nat.cast_add,
Nat.add_right_comm m n 1, Nat.add_assoc, Nat.add_comm]
#align int.cast_add Int.cast_addₓ
-- type had `HasLiftT`
@[simp, norm_cast]
theorem cast_sub (m n) : ((m - n : ℤ) : R) = m - n := by
|
simp [Int.sub_eq_add_neg, sub_eq_add_neg, Int.cast_neg, Int.cast_add]
|
@[simp, norm_cast]
theorem cast_sub (m n) : ((m - n : ℤ) : R) = m - n := by
|
Mathlib.Data.Int.Cast.Basic.122_0.3MsWc9B5PAFbTbn
|
@[simp, norm_cast]
theorem cast_sub (m n) : ((m - n : ℤ) : R) = m - n
|
Mathlib_Data_Int_Cast_Basic
|
R : Type u
inst✝ : AddGroupWithOne R
n : ℤ
⊢ ↑(bit1 n) = bit1 ↑n
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Cast of integers (additional theorems)
This file proves additional properties about the *canonical* homomorphism from
the integers into an additive group with a one (`Int.cast`).
There is also `Data.Int.Cast.Lemmas`,
which includes lemmas stated in terms of algebraic homomorphisms,
and results involving the order structure of `ℤ`.
By contrast, this file's only import beyond `Data.Int.Cast.Defs` is `Algebra.Group.Basic`.
-/
universe u
namespace Nat
variable {R : Type u} [AddGroupWithOne R]
@[simp, norm_cast]
theorem cast_sub {m n} (h : m ≤ n) : ((n - m : ℕ) : R) = n - m :=
eq_sub_of_add_eq <| by rw [← cast_add, Nat.sub_add_cancel h]
#align nat.cast_sub Nat.cast_subₓ
-- `HasLiftT` appeared in the type signature
@[simp, norm_cast]
theorem cast_pred : ∀ {n}, 0 < n → ((n - 1 : ℕ) : R) = n - 1
| 0, h => by cases h
| n + 1, _ => by rw [cast_succ, add_sub_cancel]; rfl
#align nat.cast_pred Nat.cast_pred
end Nat
open Nat
namespace Int
variable {R : Type u} [AddGroupWithOne R]
@[simp, norm_cast squash]
theorem cast_negSucc (n : ℕ) : (-[n+1] : R) = -(n + 1 : ℕ) :=
AddGroupWithOne.intCast_negSucc n
#align int.cast_neg_succ_of_nat Int.cast_negSuccₓ
-- expected `n` to be implicit, and `HasLiftT`
@[simp, norm_cast]
theorem cast_zero : ((0 : ℤ) : R) = 0 :=
(AddGroupWithOne.intCast_ofNat 0).trans Nat.cast_zero
#align int.cast_zero Int.cast_zeroₓ
-- type had `HasLiftT`
@[simp high, nolint simpNF, norm_cast] -- this lemma competes with `Int.ofNat_eq_cast` to come later
theorem cast_ofNat (n : ℕ) : ((n : ℤ) : R) = n :=
AddGroupWithOne.intCast_ofNat _
#align int.cast_coe_nat Int.cast_ofNatₓ
-- expected `n` to be implicit, and `HasLiftT`
#align int.cast_of_nat Int.cast_ofNatₓ
-- See note [no_index around OfNat.ofNat]
@[simp, norm_cast]
theorem int_cast_ofNat (n : ℕ) [n.AtLeastTwo] :
((no_index (OfNat.ofNat n) : ℤ) : R) = OfNat.ofNat n := by
simpa only [OfNat.ofNat] using AddGroupWithOne.intCast_ofNat (R := R) n
@[simp, norm_cast]
theorem cast_one : ((1 : ℤ) : R) = 1 := by
erw [cast_ofNat, Nat.cast_one]
#align int.cast_one Int.cast_oneₓ
-- type had `HasLiftT`
@[simp, norm_cast]
theorem cast_neg : ∀ n, ((-n : ℤ) : R) = -n
| (0 : ℕ) => by erw [cast_zero, neg_zero]
| (n + 1 : ℕ) => by erw [cast_ofNat, cast_negSucc]
| -[n+1] => by erw [cast_ofNat, cast_negSucc, neg_neg]
#align int.cast_neg Int.cast_negₓ
-- type had `HasLiftT`
@[simp, norm_cast]
theorem cast_subNatNat (m n) : ((Int.subNatNat m n : ℤ) : R) = m - n := by
unfold subNatNat
cases e : n - m
· simp only [ofNat_eq_coe]
simp [e, Nat.le_of_sub_eq_zero e]
· rw [cast_negSucc, Nat.add_one, ← e, Nat.cast_sub <| _root_.le_of_lt <| Nat.lt_of_sub_eq_succ e,
neg_sub]
#align int.cast_sub_nat_nat Int.cast_subNatNatₓ
-- type had `HasLiftT`
#align int.neg_of_nat_eq Int.negOfNat_eq
@[simp]
theorem cast_negOfNat (n : ℕ) : ((negOfNat n : ℤ) : R) = -n := by simp [Int.cast_neg, negOfNat_eq]
#align int.cast_neg_of_nat Int.cast_negOfNat
@[simp, norm_cast]
theorem cast_add : ∀ m n, ((m + n : ℤ) : R) = m + n
| (m : ℕ), (n : ℕ) => by simp [-Int.natCast_add, ← Int.ofNat_add]
| (m : ℕ), -[n+1] => by erw [cast_subNatNat, cast_ofNat, cast_negSucc, sub_eq_add_neg]
| -[m+1], (n : ℕ) => by
erw [cast_subNatNat, cast_ofNat, cast_negSucc, sub_eq_iff_eq_add, add_assoc,
eq_neg_add_iff_add_eq, ← Nat.cast_add, ← Nat.cast_add, Nat.add_comm]
| -[m+1], -[n+1] =>
show (-[m + n + 1+1] : R) = _ by
rw [cast_negSucc, cast_negSucc, cast_negSucc, ← neg_add_rev, ← Nat.cast_add,
Nat.add_right_comm m n 1, Nat.add_assoc, Nat.add_comm]
#align int.cast_add Int.cast_addₓ
-- type had `HasLiftT`
@[simp, norm_cast]
theorem cast_sub (m n) : ((m - n : ℤ) : R) = m - n := by
simp [Int.sub_eq_add_neg, sub_eq_add_neg, Int.cast_neg, Int.cast_add]
#align int.cast_sub Int.cast_subₓ
-- type had `HasLiftT`
section deprecated
set_option linter.deprecated false
@[norm_cast, deprecated]
theorem ofNat_bit0 (n : ℕ) : (↑(bit0 n) : ℤ) = bit0 ↑n :=
rfl
#align int.coe_nat_bit0 Int.ofNat_bit0
@[norm_cast, deprecated]
theorem ofNat_bit1 (n : ℕ) : (↑(bit1 n) : ℤ) = bit1 ↑n :=
rfl
#align int.coe_nat_bit1 Int.ofNat_bit1
@[norm_cast, deprecated]
theorem cast_bit0 (n : ℤ) : ((bit0 n : ℤ) : R) = bit0 (n : R) :=
Int.cast_add _ _
#align int.cast_bit0 Int.cast_bit0
@[norm_cast, deprecated]
theorem cast_bit1 (n : ℤ) : ((bit1 n : ℤ) : R) = bit1 (n : R) :=
by
|
rw [bit1, Int.cast_add, Int.cast_one, cast_bit0]
|
@[norm_cast, deprecated]
theorem cast_bit1 (n : ℤ) : ((bit1 n : ℤ) : R) = bit1 (n : R) :=
by
|
Mathlib.Data.Int.Cast.Basic.146_0.3MsWc9B5PAFbTbn
|
@[norm_cast, deprecated]
theorem cast_bit1 (n : ℤ) : ((bit1 n : ℤ) : R) = bit1 (n : R)
|
Mathlib_Data_Int_Cast_Basic
|
R : Type u
inst✝ : AddGroupWithOne R
n : ℤ
⊢ bit0 ↑n + 1 = bit1 ↑n
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Cast of integers (additional theorems)
This file proves additional properties about the *canonical* homomorphism from
the integers into an additive group with a one (`Int.cast`).
There is also `Data.Int.Cast.Lemmas`,
which includes lemmas stated in terms of algebraic homomorphisms,
and results involving the order structure of `ℤ`.
By contrast, this file's only import beyond `Data.Int.Cast.Defs` is `Algebra.Group.Basic`.
-/
universe u
namespace Nat
variable {R : Type u} [AddGroupWithOne R]
@[simp, norm_cast]
theorem cast_sub {m n} (h : m ≤ n) : ((n - m : ℕ) : R) = n - m :=
eq_sub_of_add_eq <| by rw [← cast_add, Nat.sub_add_cancel h]
#align nat.cast_sub Nat.cast_subₓ
-- `HasLiftT` appeared in the type signature
@[simp, norm_cast]
theorem cast_pred : ∀ {n}, 0 < n → ((n - 1 : ℕ) : R) = n - 1
| 0, h => by cases h
| n + 1, _ => by rw [cast_succ, add_sub_cancel]; rfl
#align nat.cast_pred Nat.cast_pred
end Nat
open Nat
namespace Int
variable {R : Type u} [AddGroupWithOne R]
@[simp, norm_cast squash]
theorem cast_negSucc (n : ℕ) : (-[n+1] : R) = -(n + 1 : ℕ) :=
AddGroupWithOne.intCast_negSucc n
#align int.cast_neg_succ_of_nat Int.cast_negSuccₓ
-- expected `n` to be implicit, and `HasLiftT`
@[simp, norm_cast]
theorem cast_zero : ((0 : ℤ) : R) = 0 :=
(AddGroupWithOne.intCast_ofNat 0).trans Nat.cast_zero
#align int.cast_zero Int.cast_zeroₓ
-- type had `HasLiftT`
@[simp high, nolint simpNF, norm_cast] -- this lemma competes with `Int.ofNat_eq_cast` to come later
theorem cast_ofNat (n : ℕ) : ((n : ℤ) : R) = n :=
AddGroupWithOne.intCast_ofNat _
#align int.cast_coe_nat Int.cast_ofNatₓ
-- expected `n` to be implicit, and `HasLiftT`
#align int.cast_of_nat Int.cast_ofNatₓ
-- See note [no_index around OfNat.ofNat]
@[simp, norm_cast]
theorem int_cast_ofNat (n : ℕ) [n.AtLeastTwo] :
((no_index (OfNat.ofNat n) : ℤ) : R) = OfNat.ofNat n := by
simpa only [OfNat.ofNat] using AddGroupWithOne.intCast_ofNat (R := R) n
@[simp, norm_cast]
theorem cast_one : ((1 : ℤ) : R) = 1 := by
erw [cast_ofNat, Nat.cast_one]
#align int.cast_one Int.cast_oneₓ
-- type had `HasLiftT`
@[simp, norm_cast]
theorem cast_neg : ∀ n, ((-n : ℤ) : R) = -n
| (0 : ℕ) => by erw [cast_zero, neg_zero]
| (n + 1 : ℕ) => by erw [cast_ofNat, cast_negSucc]
| -[n+1] => by erw [cast_ofNat, cast_negSucc, neg_neg]
#align int.cast_neg Int.cast_negₓ
-- type had `HasLiftT`
@[simp, norm_cast]
theorem cast_subNatNat (m n) : ((Int.subNatNat m n : ℤ) : R) = m - n := by
unfold subNatNat
cases e : n - m
· simp only [ofNat_eq_coe]
simp [e, Nat.le_of_sub_eq_zero e]
· rw [cast_negSucc, Nat.add_one, ← e, Nat.cast_sub <| _root_.le_of_lt <| Nat.lt_of_sub_eq_succ e,
neg_sub]
#align int.cast_sub_nat_nat Int.cast_subNatNatₓ
-- type had `HasLiftT`
#align int.neg_of_nat_eq Int.negOfNat_eq
@[simp]
theorem cast_negOfNat (n : ℕ) : ((negOfNat n : ℤ) : R) = -n := by simp [Int.cast_neg, negOfNat_eq]
#align int.cast_neg_of_nat Int.cast_negOfNat
@[simp, norm_cast]
theorem cast_add : ∀ m n, ((m + n : ℤ) : R) = m + n
| (m : ℕ), (n : ℕ) => by simp [-Int.natCast_add, ← Int.ofNat_add]
| (m : ℕ), -[n+1] => by erw [cast_subNatNat, cast_ofNat, cast_negSucc, sub_eq_add_neg]
| -[m+1], (n : ℕ) => by
erw [cast_subNatNat, cast_ofNat, cast_negSucc, sub_eq_iff_eq_add, add_assoc,
eq_neg_add_iff_add_eq, ← Nat.cast_add, ← Nat.cast_add, Nat.add_comm]
| -[m+1], -[n+1] =>
show (-[m + n + 1+1] : R) = _ by
rw [cast_negSucc, cast_negSucc, cast_negSucc, ← neg_add_rev, ← Nat.cast_add,
Nat.add_right_comm m n 1, Nat.add_assoc, Nat.add_comm]
#align int.cast_add Int.cast_addₓ
-- type had `HasLiftT`
@[simp, norm_cast]
theorem cast_sub (m n) : ((m - n : ℤ) : R) = m - n := by
simp [Int.sub_eq_add_neg, sub_eq_add_neg, Int.cast_neg, Int.cast_add]
#align int.cast_sub Int.cast_subₓ
-- type had `HasLiftT`
section deprecated
set_option linter.deprecated false
@[norm_cast, deprecated]
theorem ofNat_bit0 (n : ℕ) : (↑(bit0 n) : ℤ) = bit0 ↑n :=
rfl
#align int.coe_nat_bit0 Int.ofNat_bit0
@[norm_cast, deprecated]
theorem ofNat_bit1 (n : ℕ) : (↑(bit1 n) : ℤ) = bit1 ↑n :=
rfl
#align int.coe_nat_bit1 Int.ofNat_bit1
@[norm_cast, deprecated]
theorem cast_bit0 (n : ℤ) : ((bit0 n : ℤ) : R) = bit0 (n : R) :=
Int.cast_add _ _
#align int.cast_bit0 Int.cast_bit0
@[norm_cast, deprecated]
theorem cast_bit1 (n : ℤ) : ((bit1 n : ℤ) : R) = bit1 (n : R) :=
by rw [bit1, Int.cast_add, Int.cast_one, cast_bit0];
|
rfl
|
@[norm_cast, deprecated]
theorem cast_bit1 (n : ℤ) : ((bit1 n : ℤ) : R) = bit1 (n : R) :=
by rw [bit1, Int.cast_add, Int.cast_one, cast_bit0];
|
Mathlib.Data.Int.Cast.Basic.146_0.3MsWc9B5PAFbTbn
|
@[norm_cast, deprecated]
theorem cast_bit1 (n : ℤ) : ((bit1 n : ℤ) : R) = bit1 (n : R)
|
Mathlib_Data_Int_Cast_Basic
|
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