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+{"text":"\\section{Introduction}\nThe teleportation process, proposed by Bennett et al \n\\cite{1}, allows to transmit an unknown state of \na quantum system from a sender, traditionally named Alice, \nto receiver, or Bob, both are spatially separated.\nFor teleporting of a two-state particle, or qubit, \nit needs an EPR pair and a usual communication channel.\nThe large number of versions using two-particle \nentanglement has been considered \\cite{2}.\nQuantum teleportation of the photon polarized \\cite{4} and   \na single coherent mode of field \\cite{40} has been demonstrated in optical \nexperiments.\n\nAs a source of EPR pair light can be used, particularly\nthe light of the optical parametric oscillator or down \nconversion as in \\cite{40}. \nHowever the physical nature of the particles may be \ndifferent, for instance one can choose the EPR-correlated atoms \nrealized experimentally in \\cite{41}. Indeed, \nthe particles of different nature are introduced \nin the scheme called the inter-space teleportation \\cite{42},\nwhere quantum state is transferred, for example, between the atom \nand light \\cite{43}.   \n\nIn this work we consider teleportation of an entangled pair  \nto two distant parties Bob and Claire with the use of the \ntriplet in the Greenberger-Horne-Zeilinger state (GHZ). Indeed\nthe GHZ triplet has been realized experimentally \\cite{5}.\nThe main problem is to find the three-particle projection \nbasis for \na joint measurement. By contrast the single qubit state \nteleportation, the maximally entangled basis does not \naccomplish the task. The obtained basis consist of a set of\nthe three-particle projection operators with the \nmaximally entanglement of two particles only.\nMeasuring allows both receivers to recover \nan unknown state of EPR pair, but each of them \ncan not do it  separately. As it has been shown in ref. \\cite{6},\nwhere teleportation of a single qubit using \nthe GHZ triplet has been considered,  \nonly one of the receivers and not both can recovered an unknown state.\nOur results are generalized for the \nteleportation of the N-particle entanglement as the EPR-nplet.\n\nOur work is organized as follows. In section 2 the main \nfeatures of the teleportation of a single qubit are given.\nThe initial states of the entangled pair and triplet are discussed \nin section 3. In section 4 \nthe basis for the joint measurement is found.\nThe teleportation protocol and network are presented in section 5, \nwhere the results are generalized for the N- particle entanglement.\n\n\\section{Teleportation}\nThe teleportation of an unknown quantum state between \ntwo parties spatially separated, Alice and Bob, includes the following \nsteeps \\cite{1}.\nLet Alice has a two level system or qubit prepared in an unknown \nstate \n\\begin{equation}\n|\\psi_{1}\\rangle=\\alpha|0\\rangle +\\beta|1\\rangle\n\\label{01}\n\\end{equation}\nwhere $|\\alpha|^{2}+|\\beta|^{2}=1$. \nLet Alice and Bob share a maximally entangled EPR pair\n$|\\Psi_{23}\\rangle =(|01\\rangle +|10\\rangle )\/\\sqrt{2}$, so that qubit 2 \nis for Alice and qubit 3 is for Bob.\nFirst, Alice performs a joint measurement of qubits 1 and 2 in the \nBell basis consisting of four projectors \n$\\Pi_{k}=|\\pi_{k}\\rangle \\langle \\pi_{k}|$, $k=1,\\dots 4$, \n$|\\pi_{1}\\rangle =|\\Phi ^{+}_{12}\\rangle$,\n$|\\pi_{2}\\rangle =|\\Phi ^{-}_{12}\\rangle$,\n$|\\pi_{3}\\rangle =|\\Psi ^{+}_{12}\\rangle$,\n$|\\pi_{4}\\rangle =|\\Psi ^{-}_{12}\\rangle$,\nwhere the Bell states are the maximally entanglement \nof two particles  \n\\begin{equation}\n|\\Phi ^{\\pm}\\rangle =\\frac{1}{\\sqrt{2}}(|00\\rangle \\pm |11\\rangle )\n\\label{1}\n\\end{equation}\n\\begin{equation}\n|\\Psi ^{\\pm}\\rangle =\\frac{1}{\\sqrt{2}}(|01\\rangle \\pm |10\\rangle )\n\\label{2}\n\\end{equation}\n\nAs result of the joint measurement, the density operator of the \ncombined system \n$\\rho=|\\psi_{1}\\rangle \\langle \\psi_{1}|\n\\otimes |\\Psi_{23}\\rangle \\langle \\Psi_{23}|$, \nto be defined in the three-particle Hilbert space\n$H_{1}\\otimes H_{2}\\otimes H_{3}$, \nis projected into one of four Bell states.\nTwo point are important in this procedure, first, the k-th outcome \ndepends not on $\\psi_{1}$\nsecond, the reduced density matrix of the qubit 3 \n$\\rho_{3}(k)=Sp_{12}\\{\\Pi_{k}\\rho\\Pi_{k}^{\\dagger}\\}$ \nand unknown state both are connected by the unitary \ntransformation $U_{k}$\n\\begin{equation}\n\\rho_{3}(k)=U_{k}\\tilde \\rho_{1} U_{k}^{\\dagger}\n\\label{3}\n\\end{equation}\nwhere\n$\\tilde \\rho_{1}$ is the density operator of $H_{3}$, \nthat is the counterpart state\n$\\rho_{1}=|\\psi_{1}\\rangle \\langle \\psi_{1}|$, $U_{k}$ is the \nset of the Pauli matrices \n$U_{1}=\\sigma_{x}, U_{2}=-i\\sigma_{y}, U_{3}=1, \nU_{4}=\\sigma_{z}$.  \nFinally Alice sends the outcomes of her measurement to Bob \nwho performs on his qubit 3 one of four unitary operations, \ncorresponding Alice' message and has his qubit in the \noriginal state    \n$\\psi_{1}$. Teleportation is achieved.\n\n\\section{Initial states}\n\nTo teleport an EPR pair it needs a maximally entanglement of \nthree particles. From this fact let consider what initial \nstates would be used.\n\nThe wave function of an entangled pair can be chosen  as\n\\begin{equation}\n|\\Psi_{12}\\rangle = \\alpha |00\\rangle +\\beta |11\\rangle\n\\label{0001}\n\\end{equation}\nwhere $|\\alpha|^{2}+|\\beta|^{2}=1$, or in the form of EPR-pair  \n\\begin{equation}\n|\\Psi_{EPR}\\rangle = \\alpha |01\\rangle +\\beta |10\\rangle\n\\label{0002}\n\\end{equation}\nIt is possible to point eight states where three particles are maximally entangled. \nThey are\n\\begin{eqnarray}\n(|000\\rangle \\pm |111\\rangle )\/\\sqrt{2},&\\quad& (|001\\rangle \\pm |110\\rangle )\/\\sqrt{2},\n\\nonumber\n\\\\\n(|010\\rangle \\pm |101\\rangle )\/\\sqrt{2},&\\quad& (|100\\rangle \\pm |011\\rangle )\/\\sqrt{2}\n\\label{103}\n\\end{eqnarray}\nFrom the presented set of the initial states of particles without \nloss of generality we choose\n(\\ref{0002}) and triplet in the form of $GHZ$\n\\begin{equation}\n|\\Psi_{GHZ}\\rangle =\\frac{1}{\\sqrt{2}}(|000\\rangle +|111\\rangle )\n\\label{102}\n\\end{equation} \nNow we shall consider the combined system prepared initially \nin the state \n\\begin{equation}\n|\\Psi\\rangle =|\\Psi_{EPR}\\rangle \\otimes |\\Psi_{GHZ}\\rangle\n\\label{100}\n\\end{equation}\n\n\\begin{figure}\n\\unitlength=1.00mm\n\\special{em:linewidth 0.4pt}\n\\linethickness{0.4pt}\n\\begin{picture}(92.67,44.67)\n\\put(20.67,41.33){\\line(1,0){16.00}}\n\\put(20.67,35.00){\\line(1,0){15.67}}\n\\put(20.33,28.33){\\line(1,0){16.33}}\n\\put(37.33,41.33){\\circle{2.00}}\n\\put(37.00,35.00){\\circle{2.00}}\n\\put(37.33,28.33){\\circle{2.00}}\n\\put(32.67,24.33){\\framebox(9.67,20.33)[cc]{}}\n\\put(20.33,21.00){\\line(1,0){36.00}}\n\\put(56.33,21.00){\\line(0,0){0.00}}\n\\put(56.67,17.33){\\framebox(7.33,6.33)[cc]{B}}\n\\put(20.00,14.33){\\line(1,0){51.67}}\n\\put(71.33,10.67){\\framebox(7.33,6.33)[cc]{C}}\n\\put(64.33,21.00){\\line(1,0){28.00}}\n\\put(78.67,14.33){\\line(1,0){14.00}}\n\\put(42.67,34.33){\\rule{18.33\\unitlength}{1.00\\unitlength}}\n\\put(61.33,33.67){\\line(0,-1){10.00}}\n\\put(61.33,33.67){\\line(5,-2){14.00}}\n\\put(75.33,28.00){\\line(0,-1){11.00}}\n\\put(11.67,21.00){\\line(1,0){11.33}}\n\\put(11.33,21.33){\\line(4,3){9.33}}\n\\put(11.33,21.00){\\line(4,-3){9.00}}\n\\put(11.67,38.00){\\line(5,2){8.67}}\n\\put(12.00,38.00){\\line(3,-1){9.33}}\n\\put(61.33,34.67){\\circle*{2.67}}\n\\put(11.00,38.00){\\circle{2.00}}\n\\put(10.33,21.33){\\circle{2.00}}\n\\put(10.00,44.00){\\makebox(0,0)[cc]{$\\Psi_{EPR}$}}\n\\put(9.67,28.67){\\makebox(0,0)[cc]{GHZ}}\n\\put(27.00,44.33){\\makebox(0,0)[cc]{1}}\n\\put(27.00,38.33){\\makebox(0,0)[cc]{2}}\n\\put(27.00,31.33){\\makebox(0,0)[cc]{3}}\n\\put(27.00,24.67){\\makebox(0,0)[cc]{4}}\n\\put(26.67,17.33){\\makebox(0,0)[cc]{5}}\n\\put(46.00,44.67){\\makebox(0,0)[cc]{A}}\n\\put(87.67,27.00){\\makebox(0,0)[cc]{$\\Psi_{EPR}$}}\n\\end{picture}\n\\caption{Teleportation of EPR pair of qubits 1 and 2 using GHZ triplet. Alice, Bob and Claire\nshare qubits 3,4 and 5 of GHZ. Alice sends outcome of a joint measurement to \nBob and Claire who recover an unknown EPR-state.}\n\\end{figure}\nThe scheme to teleport an unknown state of EPR pair of \nqubit 1 and 2 with the use of the GHZ triplet of \nqubit 3,4 and 5 is presented in fig. 1. \nHere three parties spatially separated, Alice and two receivers \nBob and Claire share the GHZ particles 3,4 and 5. \nAlice sends outcomes of her joint measurement of qubits 1,2 \nand 3 to both receivers by classical channel. To perform the joint \nmeasurement it needs \neight projection operators to form a complete set\non which the initial  wave function $|\\Psi\\rangle $\ncan be decomposed. The choice of such basis is the main moment in the solution \nof the problem. \n\n\\section{The projection basis}\n\nIt would be possible to imagine that the set of the projection \noperators $\\Pi_{k}$ for joint measurement will consist of the maximally\n entangled states (\\ref{103}). Let denote this basis as\n$\\pi_{(123)}$. However one can find that it is not true. The reason is \nthat in series expansion of the initial wave function $|\\Psi\\rangle $ \nits projections into four vectors \n$(|010\\rangle \\pm |101\\rangle)\/\\sqrt{2}$,  \n$(|100\\rangle \\pm |011\\rangle)\/\\sqrt{2}$ of the basis $\\pi_{(123)}$\nare equal to zero.\nIt is impossible to recover unknown state of EPR pair by the such \noutcomes using unitary transformation. Therefore the maximally entangled tree- particle basis does not\nsolve the task. \n\nThe basis required turns out to be composed from the states \nwhere only two particle are maximally entangled, say 1,3 or 1,2.\nHowever the pair entangled is only the necessary condition.\n\nTo consider realization of operators $\\Pi_{k}$ we introduce classification\nwhere one of tag will be number of the particles to be maximally entangled, \nsay two or three in our case. As all complete set of vectors are \nconnected among themselves by unitary transformation \none can take an initial basis. Let the initial basis be $\\pi_{123}$\n\\begin{equation}\n|\\pi_{123}\\rangle =|ijk\\rangle \\quad i,j,k =0,1\n\\label{p123}\n\\end{equation}\nwhere each of eight elements is the state of three \nindependent or non-correlated particles. Any element of the other \nbasis can be presented by a linear superposition of\n$s\\leq 8$ vectors of the set $\\pi_{123}$. Further let suppose the \nnumber s be common for the given basis and we use it for \nclassification. So it can be introduced the set \n$\\pi_{1(23)}(s)$ consisting of the maximally entanglement of \ntwo particles 2 and 3. \nFor $s=2$ it has the form \n\\begin{equation}\n|\\pi_{1(23)}(2)\\rangle =\\{|i\\rangle |\\Phi^{\\pm}_{23}\\rangle ; \n|i\\rangle |\\Psi^{\\pm}_{23}\\rangle \\} \\quad i=0,1\n\\label{p}\n\\end{equation}\nwhere each of eight vectors, for example\n$|0\\rangle |\\Phi^{\\pm}_{23}\\rangle =(|000\\rangle \\pm |011\\rangle )\/\\sqrt{2}$, \nis presented by two elements of \n$\\pi_{123}$. \nFor the case $s=4$\n\\begin{equation}\n|\\pi_{1(23)}(4)\\rangle =\\{\n|\\pi_{1}^{\\pm}\\rangle |\\Phi^{\\pm}_{23}\\rangle ;\n|\\pi_{1}^{\\pm}\\rangle |\\Psi^{\\pm}_{23}\\rangle \\}\n\\label{pp}\n\\end{equation}\nwhere pair of vectors generating a complete single-particle set\nlooks like\n\\begin{equation}\n|\\pi_{1}^{\\pm}\\rangle =\\frac{1}{\\sqrt{2}}\n(|0\\rangle \\pm \\exp(i\\varphi)|1\\rangle )\n\\label{ppp}\n\\end{equation}\n\nThe sets presented here are complete and orthogonal, however the basis\n$\\pi_{1(23)}(2)$ does not solve the problem. The reason is that \nthe outcomes of the joint measurement depend on the wave function to be \nteleported so that the unitary transformation \nthat Bob and Claire have to perform \nat their qubits will depend on an unknown state. For base \n$\\pi_{(12)3}(4)$ the situation is similar to \n$\\pi_{(123)}(2)$, where half of projections of $\\Psi$ into the \nbasis states is equal to zero.\n\nFor telepoting  EPR pair two sets are useful for which $s=4$. \nThere are $\\pi_{1(23)}(4)$ ore  $\\pi_{(13)2}(4)$, \nwhere the particles 2,3 \nor 1,3 are maximally entangled. The structure of the initial \nstate, projection basis $\\pi_{1(23)}(4)$ and the total wave function \nare presented in fig 2.  \n\\begin{figure}[ht]\n\\unitlength=1.00mm\n\\special{em:linewidth 0.4pt}\n\\linethickness{0.4pt}\n\\begin{picture}(130.00,52.00)\n\\put(11.00,43.00){\\circle{5.66}}\n\\put(11.00,31.00){\\circle{5.66}}\n\\put(24.00,29.00){\\circle{5.66}}\n\\put(41.00,37.00){\\circle{5.66}}\n\\put(41.00,17.00){\\circle{5.66}}\n\\put(11.00,43.00){\\makebox(0,0)[cc]{1}}\n\\put(11.00,31.00){\\makebox(0,0)[cc]{2}}\n\\put(24.00,29.00){\\makebox(0,0)[cc]{3}}\n\\put(41.00,37.00){\\makebox(0,0)[cc]{4}}\n\\put(41.00,17.00){\\makebox(0,0)[cc]{5}}\n\\put(3.00,8.00){\\framebox(28.00,44.00)[cc]{}}\n\\put(33.00,8.00){\\framebox(17.00,44.00)[cc]{}}\n\\put(40.00,20.00){\\line(0,1){14.00}}\n\\put(40.00,34.00){\\line(-3,-1){14.00}}\n\\put(59.00,44.00){\\circle{5.66}}\n\\put(59.00,30.00){\\circle{5.66}}\n\\put(72.00,30.00){\\circle{5.66}}\n\\put(59.00,44.00){\\makebox(0,0)[cc]{1}}\n\\put(59.00,30.00){\\makebox(0,0)[cc]{2}}\n\\put(72.00,30.00){\\makebox(0,0)[cc]{3}}\n\\put(91.00,42.00){\\circle{5.66}}\n\\put(91.00,29.00){\\circle{5.66}}\n\\put(104.00,28.00){\\circle{5.66}}\n\\put(121.00,36.00){\\circle{5.66}}\n\\put(121.00,16.00){\\circle{5.66}}\n\\put(91.00,42.00){\\makebox(0,0)[cc]{1}}\n\\put(91.00,29.00){\\makebox(0,0)[cc]{2}}\n\\put(104.00,28.00){\\makebox(0,0)[cc]{3}}\n\\put(121.00,36.00){\\makebox(0,0)[cc]{4}}\n\\put(121.00,16.00){\\makebox(0,0)[cc]{5}}\n\\put(83.00,7.00){\\framebox(28.00,44.00)[cc]{}}\n\\put(113.00,7.00){\\framebox(17.00,44.00)[cc]{}}\n\\put(121.00,19.00){\\line(0,1){14.00}}\n\\put(8.00,13.00){\\makebox(0,0)[cc]{A}}\n\\put(46.00,42.00){\\makebox(0,0)[cc]{B}}\n\\put(46.00,19.00){\\makebox(0,0)[cc]{C}}\n\\put(88.00,12.00){\\makebox(0,0)[cc]{A}}\n\\put(127.00,41.00){\\makebox(0,0)[cc]{B}}\n\\put(127.00,18.00){\\makebox(0,0)[cc]{C}}\n\\put(27.00,29.00){\\line(3,-2){13.00}}\n\\put(22.00,1.00){\\makebox(0,0)[cc]{a)}}\n\\put(68.00,1.00){\\makebox(0,0)[cc]{b)}}\n\\put(108.00,1.00){\\makebox(0,0)[cc]{c)}}\n\\put(54.00,25.00){\\framebox(24.00,25.00)[cc]{}}\n\\put(11.00,40.00){\\line(0,-1){6.00}}\n\\put(61.00,30.00){\\line(1,0){8.00}}\n\\put(94.00,28.00){\\line(1,0){7.00}}\n\\end{picture}\n\\caption{The entanglement structure of the states. \na) The initial state, where\nqubits of EPR pair 1,2 and qubits 3,4,5 of GHZ are entangled.\nb) Projection basis $\\pi_{1(23)}(4)$. \nc) The state after measuring.}\n\\end{figure}\n \n\\section{Teleportation of EPR pair}\nUsing the obtained set $\\pi_{1(23)}$, where we put \n$\\varphi =0$, the initial wave function can be rewritten as\n\\begin{eqnarray}\n|\\Psi\\rangle&=&\n|\\pi_{1}^{+}\\rangle |\\Phi^{+}_{23}\\rangle |1\\rangle +\n|\\pi_{1}^{+}\\rangle |\\Phi^{-}_{23}\\rangle |2\\rangle \n\\nonumber\n\\\\ \n&+&\n|\\pi_{1}^{-}\\rangle |\\Phi^{+}_{23}\\rangle |3\\rangle +\n|\\pi_{1}^{-}\\rangle |\\Phi^{-}_{23}\\rangle |4\\rangle \n\\nonumber\n\\\\\n&+&\n|\\pi_{1}^{+}\\rangle |\\Psi^{+}_{23}\\rangle |5\\rangle +\n|\\pi_{1}^{+}\\rangle |\\Psi^{-}_{23}\\rangle |6\\rangle \n\\nonumber\n\\\\\n&+&\n|\\pi_{1}^{-}\\rangle |\\Psi^{+}_{23}\\rangle |7\\rangle +\n|\\pi_{1}^{-}\\rangle |\\Psi^{-}_{23}\\rangle |8\\rangle \n\\label{201}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n|1,2\\rangle &=& \\beta|00\\rangle \\pm \\alpha |11\\rangle \n\\nonumber\n\\\\\n|3,4\\rangle &=&-(\\beta|00\\rangle \\mp \\alpha |11\\rangle  )\n\\nonumber\n\\\\\n|5,6\\rangle &=&\\beta|11\\rangle \\pm \\alpha |00\\rangle \n\\nonumber\n\\\\\n|7,8\\rangle &=&-(\\beta|11\\rangle \\mp \\alpha |00\\rangle  )\n\\label{203}\n\\end{eqnarray}\nthat for each outcome the reduced density matrix of qubit 4 and 5 \nIt follows from equation (\\ref{203}) \n$\\rho_{45}(k)=Sp_{123}\\{\\Pi_{k}|\\Psi\\rangle \\langle \\Psi|\\Pi_{k}\\}$ \nis connected to the density matrix of EPR pair by unitary transformation\n\\begin{equation}\n\\rho_{45}(k)=U_{k}|\\tilde\\Psi_{EPR}\\rangle \\langle \\tilde\\Psi_{EPR}|\nU^{\\dagger}_{k}\n\\quad k=1, \\dots 8\n\\label{204}\n\\end{equation}\nwhere \n$\\tilde\\Psi_{EPR}$ is the wave function of Hilbert space\n$H_{4}\\otimes H_{5}$ and counterpart of $\\Psi_{EPR}$.\nThe unitary operator from (\\ref{204}) has the form\n$U_{1}=\\sigma_{x4}\\otimes I_{5}$,\n$U_{2}=-U_{3}=i\\sigma_{y4}\\otimes I_{5}$, \n$U_{4}=-U_{1}$,\n$U_{5}=I_{4}\\otimes \\sigma_{x5}$, \n$U_{6}=-U_{7}= I_{4}\\otimes (-i\\sigma_{y5})$,\n$U_{8}=-U_{5}$. \nwhere the Pauli operators $\\sigma_{\\gamma j}$\n$\\gamma =x,y,z$\nand identity $I_{j}$  \naffect the particle $j=4,5$.\n\nTeleportation of an unknown EPR state can be reached by the following \nprotocol:\n\\begin{enumerate}\n\\item  \nAlice performs the joint measurement of qubits 1,2 and 3 \nin basis $\\pi_{1(23)}(4)$ and sends her outcomes to Bob and Claire. \n\\item \nFor outcomes $k=1 - 4$ \nBob have to rotate his qubit by local operations\n$\\sigma_{x}$, $i\\sigma_{y}$, $-i\\sigma_{y}$, $-\\sigma_{x}$\nand Claire does nothing. As result Bob and Claire has EPR pair\nin the state $\\Psi_{EPR}$.\n\\item \nTo recover an unknown EPR state for outcomes $k=5 - 6$\nBob does nothing  and Claire performs unitary transformation \n$\\sigma_{x}$, $-i\\sigma_{y}$, $i\\sigma_{y}$,\n$-\\sigma_{x}$ on her qubit.\n\\end{enumerate}\nIn the presented protocol in half of cases only one of receivers \naffects on his particle while another acts on his particle by \nunity operator or does nothing. This version is not unique \nbecause unknown state can be recovered by different way. \nFor instance, the wave vector \n$|2\\rangle $ from (\\ref{203}) \ncan be obtained by two ways \n\\begin{equation}\n\\beta |00\\rangle -\\alpha|11\\rangle \n=i\\sigma_{y4}\\otimes I_{5}\n|\\tilde\\Psi_{EPR}\\rangle \n=\\sigma_{x4}\\otimes \\sigma_{z5}\n|\\tilde\\Psi_{EPR}\\rangle \n\\label{205}\n\\end{equation}\nThe expression (\\ref{205}) means that for outcome  \n$k=2$ both receivers Bob and Claire should simultaneously affect \non their qubits (as in the above protocol) by unitary operations  \n$\\sigma_{x4}$ and $\\sigma_{z5}$ (instead of \n$i\\sigma_{y4}$ and identity operator).\nThese differences however do not change the result. The \nmain feature of the teleportation procedure considered here is \npresence of two receivers which one can not accomplish the \ntask separately.\n\\begin{figure}\n\\unitlength=1.00mm\n\\special{em:linewidth 0.4pt}\n\\linethickness{0.4pt}\n\\begin{picture}(147.67,91.00)\n\\put(7.67,78.00){\\line(1,0){20.33}}\n\\put(136.67,77.33){\\circle{5.37}}\n\\put(32.33,77.34){\\circle*{1.33}}\n\\put(32.99,62.34){\\circle{5.37}}\n\\put(7.33,62.67){\\line(1,0){18.33}}\n\\put(46.67,57.00){\\framebox(9.00,10.00)[cc]{H}}\n\\put(55.67,63.00){\\line(1,0){9.67}}\n\\put(65.00,57.33){\\framebox(9.00,10.00)[cc]{H}}\n\\put(7.33,47.67){\\line(1,0){9.67}}\n\\put(17.33,42.00){\\framebox(9.00,10.00)[cc]{H}}\n\\put(60.33,63.00){\\circle*{1.33}}\n\\put(60.33,47.67){\\circle{5.37}}\n\\put(26.33,47.67){\\line(1,0){31.33}}\n\\put(63.33,47.67){\\line(1,0){75.67}}\n\\put(74.00,62.67){\\line(1,0){65.33}}\n\\put(32.67,76.67){\\line(0,-1){12.00}}\n\\put(60.33,62.67){\\line(0,-1){12.67}}\n\\put(30.00,47.33){\\circle*{1.33}}\n\\put(40.00,47.67){\\circle*{1.33}}\n\\put(30.00,33.33){\\circle{5.37}}\n\\put(7.33,33.33){\\line(1,0){20.00}}\n\\put(40.00,17.67){\\circle{5.37}}\n\\put(7.00,17.67){\\line(1,0){30.33}}\n\\put(30.00,46.33){\\line(0,-1){10.33}}\n\\put(40.33,47.00){\\line(0,-1){26.67}}\n\\put(69.33,47.00){\\circle*{1.33}}\n\\put(69.33,33.33){\\circle{5.37}}\n\\put(32.67,33.33){\\line(1,0){34.00}}\n\\put(76.00,28.33){\\framebox(9.00,10.00)[cc]{H}}\n\\put(90.33,34.00){\\circle{5.37}}\n\\put(123.67,12.67){\\framebox(9.00,10.00)[cc]{H}}\n\\put(118.00,33.33){\\circle{5.37}}\n\\put(126.33,33.67){\\circle{5.37}}\n\\put(137.00,17.67){\\circle*{1.33}}\n\\put(118.33,17.67){\\circle*{1.33}}\n\\put(96.67,28.00){\\framebox(9.00,10.00)[cc]{H}}\n\\put(110.33,48.00){\\circle*{1.33}}\n\\put(110.33,17.33){\\circle{5.37}}\n\\put(128.00,78.00){\\line(1,0){6.00}}\n\\put(139.33,78.00){\\line(1,0){8.00}}\n\\put(139.33,62.67){\\line(1,0){7.67}}\n\\put(139.33,47.67){\\line(1,0){8.00}}\n\\put(147.33,47.67){\\line(0,0){0.00}}\n\\put(72.00,33.33){\\line(1,0){4.33}}\n\\put(85.00,33.33){\\line(1,0){3.33}}\n\\put(93.00,33.33){\\line(1,0){4.00}}\n\\put(105.67,33.33){\\line(1,0){9.33}}\n\\put(121.00,33.33){\\line(1,0){2.67}}\n\\put(129.33,33.33){\\line(1,0){18.33}}\n\\put(42.67,17.67){\\line(1,0){65.00}}\n\\put(112.67,17.67){\\line(1,0){5.33}}\n\\put(132.33,17.67){\\line(1,0){15.33}}\n\\put(119.00,17.67){\\line(1,0){4.67}}\n\\put(126.00,78.00){\\circle*{1.33}}\n\\put(90.33,62.67){\\circle*{1.33}}\n\\put(69.33,47.00){\\line(0,-1){11.33}}\n\\put(90.33,62.33){\\line(0,-1){25.33}}\n\\put(110.33,47.67){\\line(0,-1){27.33}}\n\\put(118.33,30.33){\\line(0,-1){13.00}}\n\\put(126.33,77.67){\\line(0,-1){41.33}}\n\\put(137.00,74.33){\\line(0,-1){56.67}}\n\\put(30.33,33.00){\\makebox(0,0)[cc]{$\\times$}}\n\\put(40.33,17.33){\\makebox(0,0)[cc]{$\\times$}}\n\\put(60.67,47.67){\\makebox(0,0)[cc]{$\\times$}}\n\\put(69.33,33.33){\\makebox(0,0)[cc]{$\\times$}}\n\\put(90.33,34.00){\\makebox(0,0)[cc]{$\\times$}}\n\\put(110.33,17.33){\\makebox(0,0)[cc]{$\\times$}}\n\\put(118.33,33.33){\\makebox(0,0)[cc]{$\\times$}}\n\\put(126.33,33.67){\\makebox(0,0)[cc]{$\\times$}}\n\\put(136.67,77.33){\\makebox(0,0)[cc]{$\\times$}}\n\\put(8.00,80.00){\\makebox(0,0)[lc]{$1$}}\n\\put(11.33,79.00){\\makebox(0,0)[lb]{$\\alpha|0\\rangle+\\beta|1\\rangle$}}\n\\put(27.67,78.00){\\line(1,0){98.67}}\n\\put(43.00,62.67){\\line(1,0){3.33}}\n\\put(7.67,65.33){\\makebox(0,0)[lc]{2}}\n\\put(7.67,50.33){\\makebox(0,0)[lc]{3}}\n\\put(7.67,36.00){\\makebox(0,0)[lc]{4}}\n\\put(7.67,20.33){\\makebox(0,0)[lc]{5}}\n\\put(14.00,66.67){\\makebox(0,0)[cb]{$|1\\rangle$}}\n\\put(12.67,51.33){\\makebox(0,0)[cb]{$|0\\rangle$}}\n\\put(13.00,36.67){\\makebox(0,0)[cb]{$|0\\rangle$}}\n\\put(13.33,20.67){\\makebox(0,0)[cb]{$|0\\rangle$}}\n\\put(142.67,81.67){\\makebox(0,0)[cb]{$|1\\rangle$}}\n\\put(147.34,66.34){\\makebox(0,0)[cb]{$|0\\rangle+|1\\rangle$}}\n\\put(147.67,51.01){\\makebox(0,0)[cb]{$|0\\rangle+|1\\rangle$}}\n\\put(24.67,62.67){\\line(1,0){5.67}}\n\\put(35.67,62.67){\\line(1,0){11.00}}\n\\put(32.67,62.33){\\makebox(0,0)[cc]{$\\times$}}\n\\put(5.00,57.67){\\framebox(33.00,33.33)[cc]{}}\n\\put(22.00,87.67){\\makebox(0,0)[cc]{$EPR$}}\n\\put(5.00,5.00){\\framebox(39.33,51.00)[cc]{}}\n\\put(20.00,9.00){\\makebox(0,0)[cc]{$GHZ$}}\n\\put(126.33,78.00){\\line(1,0){3.00}}\n\\put(146.00,26.00){\\makebox(0,0)[cc]{$|\\Psi_{EPR}\\rangle$}}\n\\end{picture}\n\\caption{Network for teleportation of EPR pair}\n\\end{figure}\n\nThe network presented in fig. 3 illustrates the teleportation \nprocedure of an EPR state. It is built similarly to the  \none-particle teleportation \\cite{7}, \nand includes set of logical operations C-NOT (controlled- not) and \nHadamard transformation H. In the unit $EPR$ operation C-NOT $C_{12}$\nacting qubit 1,2 produces the entanglement of particles 1,2 of the \nform of EPR state.\n$C_{12}$ flips the second qubit (target) if and only if \nthe first (control) is logical 1.  \nThe unit  $GHZ$ prepares three-particle entanglement by the Hadamard \ntransformation H acting qubit 3 \n($H|0\\rangle=(|0\\rangle +|1\\rangle )\/\\sqrt{2}$, \n$H|1\\rangle=(|0\\rangle -|1\\rangle )\/\\sqrt{2})$ \nand two operations c-NOT \n$C_{34}, C_{35}$. On the end of the scheme the joint state of qubits \n4 and 5 being independent of others is turned out to be \n$\\Psi_{EPR}$. Indeed, the above network can be used for teleporting \nthe entangled pair of the form (\\ref{0001}).\n\nConsider the more general case of teleportation of the N-particle \nentanglement as EPR-nplet\n\\begin{equation}\n|\\Psi_{N}\\rangle =\\alpha |0\\rangle ^{N}\n+\\beta |1\\rangle ^{N}\n\\label{210}\n\\end{equation}\nusing $N+1$ qubits in the maximally entangled state as GHZ\n\\begin{equation}\n|\\Psi_{(N+1)}\\rangle = \\frac{1}{\\sqrt{2}}\n\\left(|0\\rangle ^{N+1} +\n|1\\rangle ^{N+1}\\right)\n\\label{211}\n\\end{equation} \nwhere $|i\\rangle ^{N}= |i\\rangle \\otimes \\dots|i\\rangle$, \n$i=0,1$.\nIn this procedure, that includes \n2N+1 qubits, a sender Alice and N receivers share the entanglement \nof the form (\\ref{211}).\nThe combined wave function defined in the Hilbert space \n$H_{1}\\otimes \\dots H_{2N+1}$ \nis product\n$|\\Psi\\rangle =|\\Psi_{N}\\rangle \\otimes |\\Psi_{(N+1)}\\rangle $. \n\nFor joint measuring of\nN+1 ($1,2,\\dots N,N+1$) particles it needs a complete set of \n$2^{N+1}$ projectors describing states of any maximally \nentangled pair $M,N+1$, $M=1,\\dots N$. \nFor $M=N$ \nthe required basis has the form\n\\begin{equation}\n\\pi_{1,...N-1(N,N+1)}(s)=\n\\{|\\pi_{1,...N-1}\\rangle \\otimes |\\Phi^{\\pm}_{N,N+1}\\rangle ;\n  |\\pi_{1,...N-1}\\rangle \\otimes |\\Psi^{\\pm}_{N,N+1}\\rangle \\}\n\\label{212}\n\\end{equation}\nwhere the particles\n$N,N+1$ are entangled,  \n$|\\pi_{1,...N-1}\\rangle $ are the vectors from \n$H_{1}\\otimes \\dots H_{N-1}$. As we note before the \npresence of the entangled pair is only \nthe necessary condition for the required basis. The sufficient \ncondition is magnitude of parameter s which one together with vectors \n$\\pi_{1,...N-1}$ \ncan be obtained by expanding the combined wave function over \nset of (\\ref{212}). It can be written as\n\\begin{eqnarray}\n|\\Psi \\rangle &=&\n\\{ P_{N-1} \\alpha |0\\rangle ^{N}\\pm Q_{N-1}\\beta |1\\rangle ^{N}\\}\n|\\pi_{1,...N-1}\\rangle |\\Phi^{\\pm}_{N,N+1}\\rangle\n\\nonumber\n\\\\\n&+&\n\\{ P_{N-1}\\alpha |1\\rangle ^{N}\\pm Q_{N-1}\\beta |0\\rangle ^{N}\\}\n|\\pi_{1,...N-1}\\rangle |\\Psi^{\\pm}_{N,N+1}\\rangle\n\\label{213}\n\\end{eqnarray}\nwhere\n\\begin{equation}\nP_{N-1}=\\langle \\pi_{1,...N-1}|0\\rangle ^{N-1}\n\\qquad\nQ_{N-1}=\\langle \\pi_{1,...N-1}|1\\rangle ^{N-1}\n\\label{214}\n\\end{equation}\nProcess teleportation needs the following condition  \n\\begin{equation}\nP_{N-1}\\neq 0, Q_{N-1}\\neq 0\n\\label{215}\n\\end{equation}\nIt means that all terms of the series expansion of $\\Psi$ \nhave to involve a linear superposition of\n$\\alpha |i\\rangle ^{N}$ and $\\beta|j\\rangle ^{N}$, $i\\neq j=0,1$, \nwhich one can be retrieved from \n(\\ref{210}) by unitary transformation not dependent from an unknown \nstate. The following set of vectors obeys \n(\\ref{215})\n\\begin{equation}\n|\\pi_{1,...N-1}\\rangle =\\{|\\pi^{\\pm}_{1}\\rangle ^{N-1}\\}\n\\label{216}\n\\end{equation}\nwhere $\\pi^{\\pm}_{1}$ is the one-particle set defined by\n(\\ref{ppp}). It can be easily established, noting that the set \n(\\ref{216}) consists of \n$2^{N-1}$ elements each of which contains two terms \n$|i\\rangle ^{N-1}, i=0,1$. \n\nFor the obtained basis defined by \n(\\ref{212}) and (\\ref{216})\nthe value $s$ is equal to $2^{N}$. Note, that all cases with \n$s<2^{N}$, where there are bases inclusive more than one pair of \nthe entangled particles (two pairs or triplet) does not accomplish \nthe task.  \n\nAs result, {\\em for teleporting an $N$-particle entangled state as EPR -nplet \nusing the $N+1$ particle entanglement it needs the set of the\nprojection operators with one pair of the maximally entangled particles. \nEach element of this set has to be presented by $2^{N}$ \nvectors corresponding the $N+1$-independent particle state}. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAlong with the increasing demand for the quality of communication service, future wireless systems are required to support a peak rate of thousands of megabits per second and service density of hundreds of multi-antenna devices per square meter \\cite{ZBM20}. To this end, a large number of wireless terminals and base stations (BSs) will be deployed for greater data traffic \\cite{WSN17}. However, these ubiquitous communication services are often accompanied by great challenges, in not only technical implementations but also environmental considerations. On the one hand, the deployment of plenty of BSs with a large number of radio frequency (RF) chains will immensely increase the power consumption and impose a great burden on the shape design of antenna arrays \\cite{ZWX17,ZZW19}. On the other hand, the surge in the number of network connections engenders the substantial growth of electromagnetic (EM) radiation that is nonnegligible to the public health \\cite{JHB20}. \n\nTo reduce the hardware cost of deploying large-scale antennas at the BS, we resort to the dynamic metasurface antenna (DMA). DMA is proposed as a brand-new concept for the realization of antenna arrays, where metamaterials are used in the aperture antenna design \\cite{SAI21,YXA21}. A simple DMA-based array is composed of several microstrips in parallel, each of which consists of a set of subwavelength and frequency selective resonant metamaterial elements \\cite{HDM13}. They are capable of tailoring the beams and processing signals in a dynamically configurable way \\cite{SIX16}. The application of DMAs enables a large number of adjustable elements, which are reconfigurable owing to the introduction of solid-state switchable components in each metamaterial, to be set in a small physical area \\cite{SDE19}. In addition, the number of RF chains required for DMA assisted communication is equal to that of microstrips, which is usually far less than that required in conventional antenna systems \\cite{YXA21}. Hence, both the physical size and the power consumption will be greatly reduced.\nInstead of a passive configurable metasurface that only reflects the signals, DMA array performs as an active transceiver that inherently implements signal processing techniques such as analog beamforming and combining \\cite{WSE21}.\nThe flexible architecture of metasurface as an active antenna array makes DMAs attractive for multiple-input multiple-output (MIMO) transceivers in future wireless networks \\cite{SAI21}. \n\nHowever, the application of DMAs usually restricts the achievable system spectral efficiency (SE) due to the reduction of the RF chains at the BS. To compensate for this defect, adopting reconfigurable intelligent surface (RIS) is proposed as an effective method to improve the system SE with low hardware complexity \\cite{ZLP21}. \nUnlike DMAs which perform as active metamaterials equipped at BS, RIS is a two-dimensional metamaterial surface composed of ultra-thin composite material layers that can programmatically reflect the incident EM waves to the desired directions \\cite{LNT18,AZW20}.\nRIS contains a plurality of reflecting elements that are usually constituted by positive-intrinsic-negative diodes to tune the phase of the incident signal in a software defined manner \\cite{CAZ21}. With the reconfigurable intelligent property, RIS can superimpose the incident waves by adjusting the phase shifts and then reflect them to the appropriate directions, which provides excellent flexibility for cellular mobile communications with complex propagating scenes \\cite{MA21,CMH21,ZDS21}. Consequently, by optimizing the phase shifts of RIS reflecting elements in the system design, it is possible to enhance the designed signal power while suppressing interference so as to improve the system SE. In addition, with low hardware footprints, RIS assisted communication has become a valuable wireless transmission strategy in the next generation network \\cite{ZLP21,CAZ21}. \n\nAs for the concern that the public is vulnerable to the increasing EM radiation, EM exposure is quantified at the user side and specified at a low level by communication regulatory agencies, which calls for new transmission strategy designs for MIMO uplink \\cite{CLL20,C04}. \nEM exposure refers to the radiation exposure generated by the propagation of EM waves, which usually comes from the power electronic devices and various kinds of artificial and natural lights \\cite{JHB20}. Recently, the swift development of wireless networks and the gradual maturity of the Internet of Things technology have made EM exposure a critical issue \\cite{AGM15}. Therefore, many government departments require that the EM radiation emitted by qualified electronics be kept at a low dose. To this end, specific absorption rate (SAR), which denotes the absorbed power per unit mass of human tissues with the unit W\/kg, has become a standard metric to measure the exposure level of the public \\cite{C04}. According to the Federal Communications Commission (FCC) standard, for wireless devices with frequencies in the range of 100 kHz to 6 GHz, the peak SAR on partial-body EM exposure should be limited to 1.6 W\/kg \\cite{SARstandard}. \nAs a time-averaged quadratic metric, SAR mainly relates to the near-field of transmitting antenna in uplink communication, where the peak value of SAR is much higher than the average \\cite{HLY13}. Then, SAR is considered to comply with the worst-case. In single antenna cases, SAR can be naturally complied for the worst-case by reducing the transmit power. However, for multi-antenna systems, it is inefficient to ensure the worst-case compliance in the same way, which brings the demand for the transmitter adaptive design that is actively satisfying different SAR constraints \\cite{YLH15,JYW22}.\n\nIn the fifth-generation (5G) cellular systems, the prevalence of wireless handsets with multiple antennas has evoked the EM aware optimization design for the transmission SE. For example, authors in \\cite{HLY14} proposed the matrix constraint form of EM radiation in the uplink transmission design with multi-antenna user terminals. Then, the SAR constrained sum-rate maximization precoding at users was investigated in \\cite{YLH17} for the uplink multiuser MIMO systems. Recently, due to the proposal of the controllable intelligent radio environment \\cite{HHA20}, RIS and DMA have been paid enormous attention to the next generation wireless networks. In \\cite{YXH21,YXN21}, the phase shifts of RIS and transmit precoding at BS were jointly optimized to obtain the maximum energy efficiency (EE) and resource efficiency of the downlink multiuser MIMO system. In addition to the RIS assisted system, which can dynamically adjust the propagation environment, DMAs can be adopted at the BS to reduce the energy consumption and implementation cost of massive antenna arrays. The impact of DMAs on the capacity of wireless systems was investigated in \\cite{SDE19,YXA21}, where the corresponding weights of DMAs at the BS are optimized to maximize SE and EE, respectively.\nNote that although most studies focus on the power allocation algorithms of RIS or DMA assisted systems under power constraints, the introduction of EM exposure constraints poses new challenges to the optimization. Furthermore, the fast time-varying channel is a common scenario in wireless communication systems, where instantaneous channel state information (CSI) is difficult to obtain and becomes outdated easily \\cite{GJL09,WMJ13}. Compared with instantaneous CSI, the statistical CSI, e.g., the spatial correlation and channel mean, is more likely to be stable for a longer period, thus bringing the lower cost for acquisition. In this case, efficient utilization of statistical CSI is promising for transmission design.\nIn addition, the CSI is usually not perfectly available in practical systems, which might degrade the transmission performance. Robust transmission design which incorporates the imperfect CSI effect is of practical interest \\cite{NCN21}.\n\nIn this paper, we investigate the EM aware SE maximization design in RIS and DMA assisted multiuser MIMO uplink transmission. Specifically, the transmit precoding, RIS phases, and DMA weights are jointly designed to maximize the system SE under both power and SAR constraints at users. We consider the practical scenario where RIS and DMAs are statically deployed, and hence full CSI between RIS and DMA is available in the considered system. On the other hand, both full CSI and partial CSI cases from users to RIS are considered in the optimization design. We intend to figure out the impact of EM exposure on the SE performance of hybrid RIS and DMA assisted systems by comparing with conventional systems, then compare our proposed algorithm with the conventional backoff algorithms. The main contributions of this paper are outlined as follows:\n\n\n\\begin{itemize}\t\n\t\\item[$\\bullet$] We investigate the hybrid RIS and DMA assisted multiuser MIMO system for practical interest, where RIS and DMA are actually complementary in wireless transmissions. In particular, RIS is adopted to dynamically adjust the propagation environment. Meanwhile, DMA is adopted as a new form of BS antennas to reduce the energy consumption and implementation cost. \n\t\\item[$\\bullet$] \n\tThe SAR constraints are taken into account to protect users from the high dose of EM radiation in hybrid RIS and DMA assisted transmissions. To address the EM aware problem, we propose a modified water-filling algorithm to optimize the transmit covariances, apply the minimum mean square error (MMSE), block coordinate descent (BCD) and minorization-maximization (MM) methods to optimize RIS phase shifts in closed form, and design the DMA weights by approaching the performance of unconstrainted DMA problems.\n\t\\item[$\\bullet$] The partial CSI case is studied in the transmission scenario. To reduce the complexity of the Monte Carlo method in dealing with partial CSI, we apply the deterministic equivalent (DE) method to asymptotically approximate SE. Then, we propose the AO-based SE maximization algorithm with the utilization of the channel eigenmode coupling matrices from users to RIS.\n\\end{itemize}\n\nThe rest of the paper is organized as follows: \\secref{sec:system model} elaborates the model of RIS and DMA assisted system, specifies the representation of EM exposure, and formulates the problem of SE maximization for two cases of available CSI. Based on the AO framework, \\secref{sec:opt_SE_design_PCSI} provides SE maximization algorithms of the optimization variables separately under full CSI scenario. \\secref{sec:opt_SE_design_SCSI} address the same problems with partial CSI. \\secref{sec:analysis} analyzes the convergence and complexity of the overall AO-base algorithm. In \\secref{sec:numerical_results}, numerical results are presented to analyze the performance of the proposed algorithms. Lastly, \\secref{sec:conclusion} concludes the study.\n\n\\emph{Notations}: Suppose matrix $\\mathbf{A} = \\mathrm{diag}\\{\\mathbf{A}_k\\}_{k=1}^{K}$ is the block diagonal matrix composed of $K$ sub-matrices, and the element on the $k$th diagonal block sequenced from the upper left is $\\mathbf{A}_k$. $\\mathbf{A}_{[1:k]}$ is the matrix obtained by truncating the first $k$ column vectors of matrix $\\mathbf{A}$. $\\mathbf{A}_{m,n}$ denotes the element located in row $m$ and column $n$ of matrix $\\mathbf{A}$. $\\mathbb{E}\\{\\cdot\\}$ means calculating the expected value. $\\mathbf{0}$ denotes zero vector. $\\mathbb{C}$, $\\mathbb{R}$, $\\mathbb{R}^+$ represent complex, real and positive real number sets,  respectively. $\\odot$ denotes the Hadamard product. $\\tr{\\cdot}$ means the trace. $(x)^+=\\max\\{x,0\\}$. $\\Re\\{\\cdot\\}$ means taking the real part of a complex number. $||\\cdot||_\\mathrm{F}^2$ is the Frobenius norm. $\\jmath=\\sqrt{-1}$ is the imaginary unit.\n\n\\section{System Model}\\label{sec:system model}\nConsider the hybrid RIS and DMA assisted multiuser MIMO uplink transmission with $K$ users in the single cell transmitting signals to a $M$-antenna BS simultaneously, as shown in \\figref{fig:RISDMAmodel}. Denote the user set as $\\mathcal{K}=\\{1,...,K\\}$, and the number of antennas for user $k\\in\\mathcal{K}$ is $N_k$. We assume that the encoded transmit signal from user $k$ is $\\mathbf{x}_k\\in\\mathbb{C}^{N_K\\times 1}$, which is zero mean and independent of signals from other users, i.e., $\\mathbb{E}\\{\\mathbf{x}_k\\}=\\mathbf{0}, \\forall k\\in\\mathcal{K}$ and $\\mathbb{E}\\{\\mathbf{x}_i\\mathbf{x}_j^H\\}=\\mathbf{0},\\forall i \\neq j \\in \\cal{K}$. Denote the covariance matrix corresponding transmit signal $\\mathbf{x}_k$ as $\\mathbf{Q}_k\\triangleq\\mathbb{E}\\{\\mathbf{x}_k\\mathbf{x}_k^H\\},\\forall k \\in \\cal{K}$. As the elements of $\\mathbf{x}_k$ are spatially correlated, $\\mathbf{Q}_k$ is essentially a non-diagnoal matrix. \n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.5\\textwidth]{Visio-RIS-DMA-model.eps}\n\t\\caption{The hybrid RIS and DMA assisted multiuser MIMO system.}\n\t\\label{fig:RISDMAmodel}\n\\end{figure}\n\n\\subsection{RIS Assised Model}\nIn \\figref{fig:RISDMAmodel}, the signals from transmitters are reflected in the channel with the deployment of RIS consisting of $N_R$ reflecting elements, each of which can tune the phase of the incident signal separately by applying the programmable controller \\cite{CMH21, AZW20}. \nAs the enormous increase of the RF weakens the diffraction and scattering effect, electromagnetic waves are prone to blockage by obstacles such as buildings in urban areas \\cite{HZA19}. In this paper, we assume the typical model that the direct channel from users to the BS is blocked, i.e., only the paths that users to the BS via RIS are considered in our system \\cite{HZA19}.\nIn addition, the cases that signals experience multiple reflections by RIS are ignored due to the tremendous path loss \\cite{PRW20,AZW20}. We suppose the channel matrix from user $k$ to RIS as $\\bH_{2,k}\\in\\mathbb{C}^{N_R\\times N_k}$, and that from the RIS to BS as $\\mathbf{H}_1\\in\\mathbb{C}^{M\\times N_R}$. Then, the received signal gathered at the BS side can be formulized as\n\\begin{align}\\label{equ:received_y1}\n\\mathbf{y}=\\sum\\limits_{k=1}^{K}\\mathbf{H}_1\\bm{\\Phi}\\bH_{2,k}\\mathbf{x}_k+\\mathbf{n} \\in \\mathbb{C}^{M\\times1},\n\\end{align}\nwhere $\\mathbf{n}$ is the additive noise following $\\mathcal{CN}(0,\\sigma^{2}\\mathbf{I}_{M})$, and $\\bm{\\Phi}=\\mathrm{diag}\\{\\phi_1,...,\\phi_{N_R}\\}$ denotes the phase shift matrix of RIS, whose diagonal elements are the reflection coefficients.\n\nSuppose that RIS can achieve total reflection, which means for any $n\\in\\{1,...,N_R\\}$, the reflection coefficients can be written as $\\phi_n=\\mathrm{e}^{\\jmath\\theta_n}$, where $\\theta_n$ is the phase shift introduced by the $n$th element of RIS. In addition, we adopt the ideal assumption that the reflecting elements can perform continuous phase shift, i.e., \\cite{WLW20}\n\\begin{align}\\label{equ:phase shift}\n\\phi_n\\in\\mathcal{F}_1\\triangleq\\{\\phi|\\phi=\\mathrm{e}^{\\jmath\\theta},\\theta\\in\\left[\\right.0,2\\pi\\left.\\right)\\},\n\\end{align}\nwhere $\\mathcal{F}_1$ denotes the feasible set of the reflection coefficients.\n\n\\subsection{DMA Assised Model}\nSuppose that the DMA array is equipped at the BS consisting of $M$ metamaterial units. These DMAs are composed of $S$ microstrips, each of which contains $L$ metamaterial units, i.e., $M = S \\cdot L$. In practice, DMA array can be regarded as a two-dimensional antenna array composed of a set of one-dimensional microstrips, and its configurable weight matrix ${\\boldsymbol\\Xi}\\in\\mathbb{C}^{S\\times M}$ can be written as \\cite{SDE19}\n\\begin{align}\\label{equ:Xi_structure}\n{\\boldsymbol\\Xi}_{s_1,(s_2-1)L+l}=\n\\left\\{\n\\begin{array}{cccc}\n\\xi_{s_1,l}\\in \\mathcal{F}_2  &s_1=s_2 \\\\\n0 &s_1\\neq s_2 \\\\\n\\end{array}\n\\right. \\in \\mathcal{F}_3^{S\\times M},\n\\end{align}\nwhere $s_1,s_2\\in\\{1,...,S\\}$, $l\\in\\{1,...,L\\}$, $\\{\\xi_{s_1,l}\\}_{\\forall s_1,l}$ are the weights \\iffalse(gains)\\fi of the DMA elements and the feasible set of weight matrices is denoted as $\\mathcal{F}_3^{S\\times M}$. Please note that $\\xi_{s_1,l}$ often satisfies certain constraints, e.g., its feasible set represented by $\\mathcal{F}_2$ may have the following forms \\cite{SYM17}:\n\\begin{itemize}\n\t\\item[(1).] $\\mathcal{F}_2= \\mathbb{C}$ for unconstrained DMA weights.\n\t\\item[(2).] $\\mathcal{F}_2=[x,y]$ where $0<x<y\\in\\mathbb{R}$ for amplitude only.\n\t\\item[(3).] $\\mathcal{F}_2=c\\cdot\\{0,1\\}$ where $c\\in\\mathbb{R}^+$ for binary amplitude.\n\t\\item[(4).] $\\mathcal{F}_2=\\left\\{\\frac{\\jmath+\\mathrm{e}^{\\jmath\\varphi}}{2}|\\varphi\\in\\left[\\right.0,2\\pi\\left.\\right)\\right\\}$\tfor Lorentzian phase.\n\\end{itemize}\nAs shown in \\figref{fig:RISDMAmodel}, the input of DMAs is comprised of the received signal $\\mathbf{y}\\in\\mathbb{C}^{S\\cdot L \\times 1}$ in \\eqref{equ:received_y1} at the BS side. Consider the case where the response of metamaterial elements is frequency flat as adopted in \\cite{ZSG21}, the process that the signal propagates inside the corresponding microstrips can be modeled as an equivalent causal filter with the finite impulse response, and the corresponding taps are expressed by \\cite{ZSG21}\n\\begin{align}\\label{equ:response_f}\nf_{s,l}=\\mathrm{e}^{-r_{s,l}(\\alpha_s+\\jmath\\beta_s)},\\ \\forall s\\in\\{1,...,S\\},\\ l\\in\\{1,...,L\\},\n\\end{align}\nwhere $r_{s,l}$ denotes the location of the $l$th meta-material unit in the $s$th microstrip, $\\alpha_s$ and $\\beta_s$ represent the waveguide attenuation coefficient and the wavenumber, respectively. Note that \\eqref{equ:response_f} is based on the assumption that the elements of each microstrip do not perturb the waveguide mode and do not interact with each other \\cite{SYM17}. Then, the mutual coupling effect inside the microstrip is weak and can be ignored for simplicity \\cite{SYM17}. \nDenote $\\mathbf{F}\\in\\mathbb{C}^{M\\times M}$ as the designed matrix form of the equivalent impulse response where $(\\mathbf{F})_{(s-1)L+l,(s-1)L+l}=f_{s,l}$. Then, the output radiation observed on DMAs can be expressed as \\cite{ZSG21}\n\\begin{align}\\label{equ:received_z}\n\\mathbf{z}={\\boldsymbol\\Xi}\\mathbf{F}\\mathbf{y} \\in \\mathbb{C}^{S\\times 1}.\n\\end{align} \nIt is worth emphasizing that the output of each microstrip is obtained as the linear combination of the radiation observed by the metamaterial elements of the corresponding microstrip.\nIn this paper, we consider the scenario that all the metamaterial elements introduce the same frequency response, i.e., $f_{s,l}=f,\\ \\forall s,l$ and $\\mathbf{F}={\\mathbf{I}_{M}\\cdot f}$, as that in \\cite{SDE19}. By combining \\eqref{equ:received_y1} into \\eqref{equ:received_z}, the input-output relationship of the system between the transmit signals and DMAs output can be written as\n\\begin{align}\\label{equ:relationship_zx}\n\\mathbf{z}=f\\sum\\limits_{k=1}^{K}{\\boldsymbol\\Xi}\\mathbf{H}_1\\bm{\\Phi}\\bH_{2,k}\\mathbf{x}_k+\\tilde{\\mathbf{n}} \\in \\mathbb{C}^{S\\times 1},\n\\end{align} \nwhere $\\tilde{\\mathbf{n}}=f{\\boldsymbol\\Xi}\\mathbf{n}$ is the equivalent channel noise.\n\n\\subsection{System SE}\nThe maximum achievable SE of the communication is related to the CSI available in the considered system. In the following, we consider two scenarios as follows:\n\\begin{itemize}\n\t\\item[\\emph{(a).}] \\emph{Full CSI}: $\\mathbf{H}_1$, $\\{\\bH_{2,k}\\}_{\\forall k}$ are available as the perfect instantaneous CSI in transmission.\n\t\\item[\\emph{(b).}] \\emph{Partial CSI}: The considered system has access to the instantaneous CSI for the RIS-to-DMAs channel $\\mathbf{H}_1$, but only the statistical CSI for the users-to-RIS channels $\\{\\bH_{2,k}\\}_{\\forall k}$.\t\n\\end{itemize}\n\nNote that the second scenario is realistic in situations that RIS-to-DMAs channel is slowly time-varying, whereas the users-to-RIS channels are rapidly time-varying \\cite{YXN21}. According to \\cite{SDE19}, the general SE model encompassing these scenarios can be expressed as\n\\begin{align}\\label{equ:SE_model}\n&{\\eta}_{\\mathrm{SE}}(\\mathbf{Q}, \\bm{\\Phi}, {\\boldsymbol\\Xi})=\\mathbb{E}_{\\{\\bH_{2,k}\\}}\\Bigg\\{\\log\\det\\Bigg(\\mathbf{I}_{S}+\\frac{1}{\\sigma^{2}}\\sum\\limits_{k=1}^{K}{\\boldsymbol\\Xi}\\cdot\\notag\\\\\n&\\mathbf{H}_1\\bm{\\Phi}\\bH_{2,k}\\mathbf{Q}_k\\bH_{2,k}^H\\bm{\\Phi}^H\\mathbf{H}_1^H{\\boldsymbol\\Xi}^H({\\boldsymbol\\Xi}\\bXi^H)^{-1}\\Bigg)\\Bigg\\}\\ \\mathrm {bps}\/\\mathrm {Hz},\n\\end{align}\nwhere $\\mathbf{Q}\\triangleq\\mathrm{diag}\\{\\mathbf{Q}_k\\}_{k=1}^{K}$ represents the aggregated covariance matrices. When perfect knowledge of CSI for all channels is available, $\\{\\bH_{2,k}\\}_{\\forall k}$ in \\eqref{equ:SE_model} become determined values and the expectation therein can be removed. On the other hand, when we consider the system SE with partial CSI, such as scenario (b), \\eqref{equ:SE_model} degenerates into the representation of ergodic achievable SE with statistical CSI of $\\{\\bH_{2,k}\\}_{\\forall k}$.\n\n\\subsection{Electromagnetic Exposure Model}\nIn practical wireless uplink communications, both the power and EM exposure level can place restrictions on the transmission rate from users to the BS \\cite{YLH17}. Generally, the constraints exerted on the power consumption are expressed as $\\tr{\\mathbf{Q}_k}\\leq P_{\\mathrm{max},k},\\ \\forall k\\in\\mathcal{K}$, where $P_{\\mathrm{max},k}$ denotes the power budget of the $k$th user. In addition, the EM exposure at the users is usually measured by the SAR, which can be modeled as a quadratic function of the transmitted signal $\\mathbf{x}_k$ \\cite{HLY13}. Typically, due to the variation of mass density and conductivity for different parts of tissues, there will be multiple SAR constraints for a single user \\cite{YLH15}. For the transmitter with multiple antennas, we can model the SAR with the time-averaged quadratic constraints given by \\cite{YLH15}\n\\begin{align}\\label{equ:SAR_ki}\n\\mathrm{SAR}_{k,i}&=\\mathbb{E}\\{\\tr{\\mathbf{x}^H_k\\mathbf{R}_{k,i}\\mathbf{x}_k}\\} \\notag\\\\\n&=\\tr{\\mathbf{R}_{k,i}\\mathbf{Q}_k}\\leq D_{k,i},\\quad i\\in \\{1,2,...,A_k\\},\n\\end{align}\nwhere $A_k$ denotes the number of SAR constraints for user $k$, $D_{k,i}$ is the maximum SAR value that the $i$th body part of the $k$th user can be exposed to EM field, and $\\mathbf{R}_{k,i}$ is the $i$th SAR matrix of user $k$ describing the radiation coefficients of transmit signals with the unit of each entry as $\\mathrm{kg}^{-1}$. Commonly, the SAR matrix is dependent on the conductivity and frequency, as well as the employed instrument that dictates the surrounding boundary conditions \\cite{LCA13}. However, work in \\cite{CCM04} shows that the SAR measurements do not vary significantly over a fairly wide range of carrier frequencies (1.8 -- 2 GHz). Note that for narrowband communication with a bandwidth less than 200 MHz, SAR measurements are almost the same for the given angle of antennas so that only a single pointwise SAR matrix is required for an orthogonal frequency division multiplexing system. Therefore, it is reasonable to assume all the sub-carriers in the systems share the same SAR matrix. \n\n\\subsection{Problem Formulation}\nIn this paper, we investigate strategy design for the RIS and DMA assisted multiuser MIMO uplink transmission with EM exposure constraints, where we aim to optimize the adjustable parameters $\\{\\mathbf{Q}_k\\}_{\\forall k}$, $\\bm{\\Phi}$ and ${\\boldsymbol\\Xi}$ to maximize the system SE. Applying the relevant constraints to \\eqref{equ:SE_model}, our work can be formulized as\n\\begin{subequations}\\label{eq:problem1}\n\t\\begin{align}\n\t\\mathcal{P}_1:\\quad\\underset{\\{\\mathbf{Q}_k\\},\\bm{\\Phi},{\\boldsymbol\\Xi}} \\max \\quad &{\\eta}_{\\mathrm{SE}}(\\mathbf{Q}, \\bm{\\Phi}, {\\boldsymbol\\Xi}), \\label{p1a}\\\\\n\t{\\mathrm{s.t.}} \\quad &\\tr{\\mathbf{Q}_k}\\leqP_{\\mathrm{max},k},\\quad \\mathbf{Q}_k \\succeq \\mathbf{0}, \\label{p1b}\\\\\n\t&\\tr{\\mathbf{R}_{k,i}\\mathbf{Q}_k}\\leq D_{k,i},\\quad \\forall k,i, \\label{p1c}\\\\\n\t&\\phi_n \\in \\mathcal{F}_1,\\quad \\forall n, \\label{p1d}\\\\\n\t&{\\boldsymbol\\Xi} \\in \\mathcal{F}_3^{S\\times M}. \\label{p1e}\n\t\\end{align}\n\\end{subequations}\nThe constraints \\eqref{p1b} and \\eqref{p1c} keep the transmission power and EM exposure level below some specified values, and \\eqref{p1d}, \\eqref{p1e} prescribe the formats of RIS phase shift matrix $\\bm{\\Phi}$ and DMA weight matrix ${\\boldsymbol\\Xi}$,  respectively. \n\nNote that it is difficult to tackle with $\\mathcal{P}_1$ directly. Firstly, due to the introduction of SAR constraints, the optimization of transmission covariance matrix $\\{\\mathbf{Q}_k\\}_{\\forall k}$ can not be transformed into the traditional power allocation problem, which brings challenges in handling $\\mathbf{Q}_k$. Secondly, calculating \\eqref{p1a} in the scenario of partial CSI will bring expensive computational cost. Thirdly, the non-convex constraints of \\eqref{p1d} and \\eqref{p1e} further complicate the joint optimization problem. With these in mind, we intend to investigate approaches to address these difficulties.\n\n\\section{EM Aware SE Optimization with Full CSI} \\label{sec:opt_SE_design_PCSI}\nWith full CSI of $\\mathbf{H}_1$ and $\\{\\bH_{2,k}\\}$, the expectation operation in \\eqref{equ:SE_model} is removed. Note that the variables of \\eqref{p1a} are tightly coupled in $\\mathcal{P}_1$. It is complicated to optimize $\\mathbf{Q}$, $\\bm{\\Phi}$, and ${\\boldsymbol\\Xi}$ jointly, especially for the case with high dimensional matrix and non-convex constraints. To reduce complexity, we apply the AO method, which is commonly used for problems with numerous optimization variables \\cite{YXN21,YXA21}, to optimize $\\mathbf{Q}$, $\\bm{\\Phi}$, and ${\\boldsymbol\\Xi}$ separately so that the problem can be handled in an alternating manner.\n\n\n\\subsection{Transmit Covariance Matrices Design}\\label{subsec:Qk1}\nSuppose $\\bm{\\Phi}$ and ${\\boldsymbol\\Xi}$ are fixed values in the feasible set that satisfy the corresponding constraints. Then \\eqref{p1d} and \\eqref{p1e} can be regarded as irrelevant constraints when independently optimizing the transmit covariance matrices. \nBy applying Sylvester's determinant theorem \\cite{Matrix}, we rewrite \\eqref{equ:SE_model} as\n\\begin{align}\\label{equ:SE_model1}\n&{\\eta}_{\\mathrm{SE}}(\\mathbf{Q})=\\log\\det\\Bigg(\\mathbf{I}_{S}+ \\notag\\\\\n&\\left.\\frac{1}{\\sigma^{2}}\\sum\\limits_{k=1}^{K}\\mathbf{H}_1\\bm{\\Phi}\\bH_{2,k}\\mathbf{Q}_k\\bH_{2,k}^H\\bm{\\Phi}^H\\mathbf{H}_1^H{\\boldsymbol\\Xi}^H({\\boldsymbol\\Xi}\\bXi^H)^{-1}{\\boldsymbol\\Xi}\\right),\n\\end{align}\nwhere ${\\boldsymbol\\Xi}^H({\\boldsymbol\\Xi}\\bXi^H)^{-1}{\\boldsymbol\\Xi}$ is the projection matrix \\cite[Ch. 5.9]{Matrix}. Denote the compact singular valued decomposition of ${\\boldsymbol\\Xi}$ as\n\\begin{align}\\label{equ:Xidecompose}\n\t{\\boldsymbol\\Xi}=\\mathbf{U}_1\\widetilde{\\bXi}\\widetilde{\\bV}_1^H,\n\\end{align}\t\nwhere $\\mathbf{U}_1\\in\\mathbb{C}^{S\\times S}$ is the unitary matrix, $\\widetilde{\\bXi}\\in\\mathbb{C}^{S\\times S}$ is the diagonal matrix with singular values in the descending order, and $\\widetilde{\\bV}_1\\in\\mathbb{C}^{M\\times S}$ is the matrix consisting of the first $S$ columns, which are mutually orthonormal, of the right singular vector matrix of ${\\boldsymbol\\Xi}$. According to \\cite{Matrix}, the projection matrix can be simplified as ${\\boldsymbol\\Xi}^H({\\boldsymbol\\Xi}\\bXi^H)^{-1}{\\boldsymbol\\Xi}=\\widetilde{\\bV}_1\\Vtildeone^H$. \nThen, \\eqref{equ:SE_model1} is equivalent to \n\\begin{align}\\label{equ:SE_model2}\n{\\eta}_{\\mathrm{SE}}(\\mathbf{Q})&=\\log\\det\\Bigg(\\mathbf{I}_{S}+\\notag\\\\\n&\\left.\\frac{1}{\\sigma^{2}}\\sum\\limits_{k=1}^{K}\\widetilde{\\bV}_1^H\\mathbf{H}_1\\bm{\\Phi}\\bH_{2,k}\\mathbf{Q}_k\\bH_{2,k}^H\\bm{\\Phi}^H\\mathbf{H}_1^H\\widetilde{\\bV}_1\\right).\n\\end{align}\nDenote $\\mathbf{G}_k=\\widetilde{\\bV}_1^H\\mathbf{H}_1\\bm{\\Phi}\\bH_{2,k}\\in\\mathbb{C}^{S\\times N_k}$ as the equivalent channel matrix, then $\\mathcal{P}_1$ for the optimization design of $\\{\\mathbf{Q}_k\\}$ with full CSI can be formulated as\n\\begin{subequations}\\label{eq:problem2a}\n\t\\begin{align}\n\t\\mathcal{P}_{2}^{a}:\\ \\underset{\\{\\mathbf{Q}_k\\}_{\\forall k}} \\max\\  &\\log\\det\\left(\\mathbf{I}_{S}+\\frac{1}{\\sigma^{2}}\\sum\\limits_{k=1}^{K}\\mathbf{G}_k\\mathbf{Q}_k\\mathbf{G}_k^H\\right), \\label{p2aa}\\\\\n\t{\\mathrm{s.t.}} \\ &\\tr{\\mathbf{Q}_k}\\leqP_{\\mathrm{max},k},\\quad \\mathbf{Q}_k \\succeq \\mathbf{0}, \\label{p2ab}\\\\\n\t&\\tr{\\mathbf{R}_{k,i}\\mathbf{Q}_k}\\leq D_{k,i},\\quad \\forall k,i. \\label{p2ac}\n\t\\end{align}\n\\end{subequations}\n\nIt is noteworthy that \\eqref{p2aa} is concave on $\\{\\mathbf{Q}_k\\}_{\\forall k}$, and \\eqref{p2ab}, \\eqref{p2ac} are linear constraints. Then, $\\mathcal{P}_{2}^{a}$ is a semidefinite programming problem. To this end, we consider its Lagrange dual function written as\n\\begin{align}\\label{equ:Lagrangian1}\n&\\mathcal{L}\\left(\\mathbf{Q},\\{\\mu_k\\},\\{\\lambda_{k,i}\\}\\right)\n={\\eta}_{\\mathrm{SE}}(\\mathbf{Q}) - \\sum\\limits_{k=1}^{K}\\mu_k(\\tr{\\mathbf{Q}_k}-P_{\\mathrm{max},k})\\notag\\\\ \n&\\quad\\quad\\ -\\sum\\limits_{k=1}^{K}\\sum\\limits_{i=1}^{A_k}\\lambda_{k,i}(\\tr{\\mathbf{R}_{k,i}\\mathbf{Q}_k}-D_{k,i}), \n\\end{align} \nwhere $\\mu_k\\geq 0, \\lambda_{k,i}\\geq 0, \\forall k,i$ are Lagrange multipliers. As elaborated in \\cite{Convex}, the strong dual problem of $\\mathcal{P}_{2}^{a}$ can be expressed as\n\\begin{align}\\label{equ:dual_problem1}\n\\underset{\\{\\mu_k\\}_{\\forall k},\\{\\lambda_{k,i}\\}_{\\forall k,i}}\\min\\ \\underset{\\{\\mathbf{Q}_k\\}_{\\forall k}}\\max \\quad \\mathcal{L}\\left(\\mathbf{Q},\\{\\mu_k\\},\\{\\lambda_{k,i}\\}\\right).\n\\end{align}\n\n\\begin{prop}\\label{Prop:Qk_optimal1}\n\tAssume the optimal dual values for power and SAR constraints of the problem \\eqref{equ:dual_problem1} are $\\{\\mu_{k,\\mathrm{opt}}\\}_{\\forall k}$ and $\\{\\lambda_{k,i,\\mathrm{opt}}\\}_{\\forall k,i}$, respectively. Denote $\\mathbf{K}_{k,\\mathrm{opt}}=\\mu_{k,\\mathrm{opt}}\\mathbf{I}_{N_k}+\\sum\\limits_{i=1}^{A_k}\\lambda_{k,i,\\mathrm{opt}}\\mathbf{R}_{k,i}$ and $\\mathbf{C}_{ipn}=\\sigma^{2}\\mathbf{I}_{S}+\\sum\\limits_{j=1\\atop j\\neq k}^{K}\\mathbf{G}_j\\mathbf{Q}_{k,\\mathrm{opt}}\\mathbf{G}_j^H$. Then, the optimal $\\mathbf{Q}_k$ can be expressed by\n\t\\begin{align}\\label{Optimal_Q1}\n\t\\mathbf{Q}_{k,\\mathrm{opt}}=\\mathbf{K}_{k,\\mathrm{opt}}^{-1\/2}\\mathbf{U}_{k}\\bm{\\Lambda}_{k}\\mathbf{U}_{k}^H\\mathbf{K}_{k,\\mathrm{opt}}^{-1\/2},\n\t\\end{align}\n\twhere $\\mathbf{U}_{k}$ is the eigenvector matrix derived by\n\t\\begin{align}\\label{Kk_Hk_eig_decomp}\n\t\\mathbf{K}_{k,\\mathrm{opt}}^{-1\/2}\\mathbf{G}_{k}^H\\mathbf{C}_{ipn}^{-1}\\mathbf{G}_k\\mathbf{K}_{k,\\mathrm{opt}}^{-1\/2}\n\t=\\mathbf{U}_k{\\boldsymbol\\Sigma}_k\\mathbf{U}_k^H.\t\n\t\\end{align}\n\tAssume that ${\\boldsymbol\\Sigma}_k=\\mathrm{diag}\\{p_{k,1},...,p_{k,N_k}\\}$ and $p_{k,1}\\geq p_{k,2}\\geq ...\\geq p_{k,N_k}$ for all $k\\in\\mathcal{K}$. Then, the optimal power allocation matrix is obtained by\n\t\\begin{align}\\label{equ:Lambdak}\n\t\\bm{\\Lambda}_{k}=\\mathrm{diag}\\{\\left(1-1\/p_{k,1}\\right)^+,...,\\left(1-1\/p_{k,N_k}\\right)^+\\}.\t\n\t\\end{align}\n\\end{prop}\nThe proof of \\emph{\\propref{Prop:Qk_optimal1}} is similar to that in \\cite[Th. 3.6]{YLH15}, thus is omitted in this paper. The EM aware transmit covariance matrices optimization with $\\bm{\\Phi}$ and ${\\boldsymbol\\Xi}$ fixed is detailed in \\textbf{\\alref{alg:waterfilling_PCSI}}.\n\\begin{algorithm}[h]\n\t\\caption{Covariance Matrices Design with Full CSI}\n\t\\label{alg:waterfilling_PCSI}\n\t\\begin{algorithmic}[1]\n\t\t\\State Initialize $\\mu_{k}^{(0)}$, $\\lambda_{k,i}^{(0)},\\forall k,i$, iteration index $\\iota=0$.\n\t\t\\Repeat\n\t\t\\State Calculate $\\mathbf{U}_k^{(\\iota)}$ and $\\bm{\\Lambda}_{k}^{(\\iota)},\\forall k$ by \\eqref{Kk_Hk_eig_decomp} and \\eqref{equ:Lambdak}.\n\t\t\\State Obtain $\\mathbf{Q}_k^{(\\iota+1)},\\forall k$ by \\eqref{Optimal_Q1}.\n\t\t\\State Set $\\iota=\\iota+1$.\n\t\t\\State Update $\\mu_{k}^{(\\iota)}$, $\\lambda_{k,i}^{(\\iota)},\\forall k,i$ by minimizing $\\mathcal{L}$ in \\eqref{equ:dual_problem1}.\n\t\t\\Until $\\{\\mu_k\\}_{\\forall k}$, $\\{\\lambda_{k,i}\\}_{\\forall k,i}$ converge.\n\t\\end{algorithmic}\n\\end{algorithm}  \n\n\n\\subsection{RIS Phase Shift Matrix Design}\\label{subsec:bPhi1}\nConsider the optimization of variable $\\bm{\\Phi}$ with $\\mathbf{Q}$ and ${\\boldsymbol\\Xi}$ being fixed. Based on the process similar to the previous section, problem $\\mathcal{P}_1$ regarding the optimization target $\\bm{\\Phi}$ is formulated as\n\\begin{subequations}\\label{eq:problem2b}\n \t\\begin{align}\n \t\\mathcal{P}_{2}^{b}:\\underset{\\bm{\\Phi}} \\max\\ &\\log\\det\\left(\\mathbf{I}_{S}+\\frac{1}{\\sigma^{2}}\\widetilde{\\bV}_1^H\\mathbf{H}_1\\bm{\\Phi}\\mathbf{P}\\bm{\\Phi}^H\\mathbf{H}_1^H\\widetilde{\\bV}_1\\right), \\label{p2ba}\\\\\n \t{\\mathrm{s.t.}} \\ &\\phi_n \\in \\mathcal{F}_1,\\quad \\forall n, \\label{p2bb}\n \t\\end{align}\n\\end{subequations}\nwhere $\\mathbf{P}=\\sum\\limits_{k=1}^{K}\\bH_{2,k}\\mathbf{Q}_k\\bH_{2,k}^H$. Because of the non-convexity on both the objective function \\eqref{p2ba} and the constraint \\eqref{p2bb}, there exist great challenges to deal with $\\mathcal{P}_{2}^{b}$ directly. Actually, it can be observed that \\eqref{p2ba} can be regarded as the SE expression of an equivalent communication system with the channel matrix $\\widetilde{\\bV}_1^H\\mathbf{H}_1\\bm{\\Phi}\\mathbf{P}^{1\/2}$. Specifically, the hypothetical input-output system is expressed as\n\\begin{align}\\label{equ:eqsystem}\n\\mathbf{y}_e=\\widetilde{\\bV}_1^H\\mathbf{H}_1\\bm{\\Phi}\\mathbf{P}^{1\/2}\\mathbf{x}_e+\\mathbf{n}_e \\in \\mathbb{C}^{S\\times1},\n\\end{align}\nwhere $\\mathbf{x}_e\\sim\\mathcal{CN}(0,\\mathbf{I}_{N_R})$, $\\mathbf{y}_e$ and $\\mathbf{n}_e\\sim\\mathcal{CN}(0,\\sigma^{2}\\mathbf{I}_{S})$ are the equivalent transmit signal, receive signal and thermal noise, respectively. Denote $\\mathbf{U}_e\\in\\mathbb{C}^{S\\times N_R}$ as the equivalent receiving matrix, the MMSE matrix of the received signal after passing the linear decoder is \n\\begin{align}\\label{equ:2MSE}\n&\\mathbf{E}_e=\\mathbb{E}\\left\\{\\left(\\mathbf{U}_e^H\\mathbf{y}_e-\\mathbf{x}_e\\right)\\left(\\mathbf{U}_e^H\\mathbf{y}_e-\\mathbf{x}_e\\right)^H\\right\\}=\\sigma^{2}\\mathbf{U}_e^H\\mathbf{U}_e+\\notag\\\\\n&(\\mathbf{U}_e^H\\widetilde{\\bV}_1^H\\mathbf{H}_1\\bm{\\Phi}\\mathbf{P}^{\\frac{1}{2}}-\\mathbf{I}_{N_R})(\\mathbf{U}_e^H\\widetilde{\\bV}_1^H\\mathbf{H}_1\\bm{\\Phi}\\mathbf{P}^{\\frac{1}{2}}-\\mathbf{I}_{N_R})^H.\n\\end{align}\nAccording to the weighted MMSE (WMMSE) method elaborated in \\cite[Th. 1]{SRL11}, $\\mathcal{P}_{2}^{b}$ is equivalent to a WMMSE minimization problem formulated by\n\\begin{subequations}\\label{eq:problem2bMMSE}\n\t\\begin{align}\n\t\\mathcal{P}_{2}^{\\mathrm{MSE}}:\\ \\underset{\\mathbf{W}_e,\\mathbf{U}_e,\\bm{\\Phi}} \\min\\  &h(\\mathbf{W}_e,\\mathbf{U}_e,\\bm{\\Phi})\\triangleq\\tr{\\mathbf{W}_e\\mathbf{E}_e}\\notag\\\\\n\t&\\qquad\\qquad-\\log\\det(\\mathbf{W}_e), \\label{pMSEa}\\\\\n\t{\\mathrm{s.t.}} \\quad &\\phi_n \\in \\mathcal{F}_1,\\quad \\forall n, \\label{pMSEb}\n\t\\end{align}\n\\end{subequations}\nwhere $\\mathbf{W}_e\\in\\mathbb{C}^{N_R\\times N_R}$ is employed as the auxiliary variable. Note that \\eqref{pMSEa} is convex on each optimization variable with the other two variables fixed. To this end, we adopt the BCD method \\cite{WGS12}, which is the generalization of AO, to minimize \\eqref{pMSEa} by iteratively updating $\\mathbf{W}_e$, $\\mathbf{U}_e$ and $\\bm{\\Phi}$ until converging. When other variables are fixed, it is clear to obtain the closed form solutions of the optimal $\\mathbf{W}_{e}$ and $\\mathbf{U}_e$, which are expressed by\n\\begin{align}\n\\label{equ:2Weopt}\n&\\mathbf{W}_{e}^{\\mathrm{opt}}=\\mathbf{E}_e^{-1},\\\\\n\\label{equ:2Ueopt}\n&\\mathbf{U}_{e}^{\\mathrm{opt}}=(\\sigma^{2}\\mathbf{I}_S+\\widetilde{\\bV}_1^H\\mathbf{H}_1\\bm{\\Phi}\\mathbf{P}\\bm{\\Phi}^H\\mathbf{H}_1^H\\widetilde{\\bV}_1)^{-1}\\widetilde{\\bV}_1^H\\mathbf{H}_1\\bm{\\Phi}\\mathbf{P}^{\\frac{1}{2}}.\n\\end{align}\nWith given $\\mathbf{W}_{e}$ and $\\mathbf{U}_e$, $\\mathcal{P}_{2}^{\\mathrm{MSE}}$ is reduced to \n\\begin{subequations}\\label{eq:problem2bMSEPhi}\n\t\\begin{align}\n\t\\mathcal{P}_{2,\\bm{\\Phi}}^{\\mathrm{MSE}}:\\ \\underset{\\bm{\\Phi}} \\min\\  &\\tr{\\bm{\\Phi}^H\\mathbf{A}\\bm{\\Phi}\\mathbf{P}}-\\tr{\\bm{\\Phi}^H\\mathbf{B}^H}-\\tr{\\bm{\\Phi}\\mathbf{B}}, \\label{pMSEpa}\\\\\n\t{\\mathrm{s.t.}}\\ &\\phi_n \\in \\mathcal{F}_1,\\quad \\forall n, \\label{pMSEpb}\n\t\\end{align}\n\\end{subequations}\nwhere $\\mathbf{A}=\\mathbf{H}_1^H\\widetilde{\\bV}_1\\mathbf{U}_e\\mathbf{W}_e\\mathbf{U}_e^H\\widetilde{\\bV}_1^H\\mathbf{H}_1\\in\\mathbb{C}^{N_R\\times N_R}$, and $\\mathbf{B}=\\mathbf{P}^{\\frac{1}{2}}\\mathbf{W}_e\\mathbf{U}_e^H\\widetilde{\\bV}_1^H\\mathbf{H}_1\\in\\mathbb{C}^{N_R\\times N_R}$. Considering that $\\bm{\\Phi}$ is a diagonal matrix where the modulus of each element is unity, we denote diagonal element vectors $\\bm{\\phi}=[\\phi_1,...,\\phi_n]^T$, and $\\mathbf{b}=\\big[\\mathbf{B}_{1,1},...,\\mathbf{B}_{N_R,N_R}\\big]^T$. Then, by employing matrix identity derivation, we get \\cite{Matrix}\n\\begin{align}\n\\label{equ:identity1}\n&\\tr{\\bm{\\Phi}^H\\mathbf{A}\\bm{\\Phi}\\mathbf{P}}=\\bm{\\phi}^H(\\mathbf{A}\\odot\\mathbf{P}^T)\\bm{\\phi},\\\\\n\\label{equ:identity23}\n&\\tr{\\bm{\\Phi}^H\\mathbf{B}^H}=\\mathbf{b}^H\\bm{\\phi}^*,\\qquad \\tr{\\bm{\\Phi}\\mathbf{B}}=\\bm{\\phi}^T\\mathbf{b}.\n\\end{align}\nTherefore, the equivalent problem of $\\mathcal{P}_{2,\\bm{\\Phi}}^{\\mathrm{MSE}}$ can be written as\n\\begin{subequations}\\label{eq:problem2bMSEPhiEQ}\n\t\\begin{align}\n\t\\overline{\\mathcal{P}}_{2,\\bm{\\phi}}^{\\mathrm{MSE}}:\\ \\underset{\\bm{\\phi}} \\min\\  &g(\\bm{\\phi})=\\bm{\\phi}^H(\\mathbf{A}\\odot\\mathbf{P}^T)\\bm{\\phi}-2\\Re\\{\\bm{\\phi}^H\\mathbf{b}^*\\}, \\label{pMSEeqa}\\\\\n\t{\\mathrm{s.t.}}\\ &\\phi_n \\in \\mathcal{F}_1,\\quad \\forall n. \\label{pMSEeqb}\n\t\\end{align}\n\\end{subequations}\n\\begin{prop}\\label{Prop:bPhi_optimal2}\n\tDenote $\\bm{\\Delta}=\\mathbf{A}\\odot\\mathbf{P}^T$ and its maximum eigenvalue as $\\lambda_{\\mathrm{max}}$. The suboptimal solution of $\\overline{\\mathcal{P}}_{2,\\bm{\\phi}}^{\\mathrm{MSE}}$ can be obtained by the iterative MM procedure, where each surrogate subproblem constructed from the previous iteration result can be written as\n\t\\begin{subequations}\\label{eq:problem2MM}\n\t\t\\begin{align}\n\t\t\\overline{\\mathcal{P}}_{2,\\bm{\\phi}}^{(\\zeta)}:\\ \\bm{\\phi}^{(\\zeta+1)}=\\argmax{\\bm{\\phi}}\\  &G(\\bm{\\phi}|\\bm{\\phi}^{(\\zeta)})=\\Re\\{\\bm{\\phi}^H\\mathbf{c}^{(\\zeta)}\\}, \\label{pMMa}\\\\\n\t\t{\\mathrm{s.t.}}\\ &|\\phi_n|=1\\quad n=1,...,N_R, \\label{pMMb}\n\t\t\\end{align}\n\t\\end{subequations}\n\twhere\n\t\\begin{align}\\label{equ:iteration_cl}\n\t\\mathbf{c}^{(\\zeta)}=(\\lambda_{\\mathrm{max}}\\mathbf{I}_{N_R}-\\bm{\\Delta})\\bm{\\phi}^{(\\zeta)}+\\mathbf{b}^*,\n\t\\end{align}\n\tand $\\zeta\\in\\mathbb{N}$ is the iteration index of MM method. Recover $\\bm{\\Phi}^{(\\zeta)}=\\mathrm{diag}\\{\\phi_1^{(\\zeta)},...,\\phi_{N_R}^{(\\zeta)}\\}$, then $\\left\\{\\bm{\\Phi}^{(\\zeta)}\\right\\}^{\\infty}_{\\zeta=0}$ will converge to the suboptimal solution of the problem \\eqref{eq:problem2bMSEPhi} where each point represents the local minimizer of the problem. \n\\end{prop}\n\\IEEEproof Refer to \\appref{app:A}.\n\nPlease notice that if we express $c_n^{(\\zeta)}=\\left|c_n^{(\\zeta)}\\right|\\mathrm{e}^{\\jmath\\vartheta_n^{(\\zeta)}}$ as the $n$th element of $\\mathbf{c}^{(\\zeta)}$, it is obvious to obtain the closed form solution of $\\bm{\\Phi}^{(\\zeta)}$ where \n\\begin{align}\\label{equ:optimal_phi}\n&\\phi_{n,\\mathrm{opt}}^{(\\zeta+1)}=\\mathrm{e}^{\\jmath\\vartheta_n^{(\\zeta)}}.\n\\end{align} \nTherefore, the RIS phase shift matrix optimization with $\\mathbf{Q}$ and ${\\boldsymbol\\Xi}$ fixed is proposed in \\textbf{\\alref{alg:WMMSE_PCSI}}.\n\\begin{algorithm}[h]\n\t\\caption{RIS Phase Shift Matrix Design with Full CSI}\n\t\\label{alg:WMMSE_PCSI}\n\t\\begin{algorithmic}[1]\n\t\t\\State Initialize $\\bm{\\Phi}^{(0)},\\mathbf{U}_e^{(0)},\\mathbf{W}_e^{(0)}$, iteration indices $t=0$, $\\zeta$, thresholds $\\epsilon_1$, $\\epsilon_2$, and calculate $h^{(0)}$ by \\eqref{pMSEa}.\n\t\t\\Repeat\n\t\t\\State Obtain $\\mathbf{W}_e^{(t+1)}$ and $\\mathbf{U}_e^{(t+1)}$ by \\eqref{equ:2Weopt} and \\eqref{equ:2Ueopt}.\n\t\t\\State Set $\\zeta=0$, $\\bm{\\Phi}^{(0)}=\\bm{\\Phi}^{(t)}$, and obtain $g(\\bm{\\phi}^{(0)})$ by \\eqref{pMSEeqa}.\n\t\t\\Repeat \n\t\t\\State Calculate $\\mathbf{c}^{(\\zeta)}$ by \\eqref{equ:iteration_cl} and express its phase.\n\t\t\\State Update $\\bm{\\phi}^{(\\zeta+1)}$ with the element obtained by \\eqref{equ:optimal_phi}.\n\t\t\\State Set $\\zeta=\\zeta+1$.\n\t\t\\State Calculate $g(\\bm{\\phi}^{(\\zeta)})$ by \\eqref{pMSEeqa}.\n\t\t\\Until $|g(\\bm{\\phi}^{(\\zeta)})-g(\\bm{\\phi}^{(\\zeta-1)})|\\leq \\epsilon_1$.\n\t\t\\State Return $\\bm{\\Phi}^{(t+1)}$ with the element $\\phi_n^{(\\zeta)}$.\n\t\t\\State Set $t=t+1$.\n\t\t\\State Update $h^{(t)}$ by \\eqref{pMSEa}.\n\t\t\\Until $|h^{(t)}-h^{(t-1)}|\\leq \\epsilon_2$.\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\n\n\\subsection{DMA Weight Matrix Design}\\label{subsec:bXi1}\nWe suppose both $\\mathbf{Q}$ and $\\bm{\\Phi}$ are fixed when optimizing the weight matrix ${\\boldsymbol\\Xi}$. Therefore, $\\mathcal{P}_1$ is degenerate into \n\\begin{subequations}\\label{eq:problem2c}\n\t\\begin{align}\n\t\\mathcal{P}_{2}^{c}:\\quad\\underset{{\\boldsymbol\\Xi}} \\max\\quad  &{\\eta}_{\\mathrm{SE}}({\\boldsymbol\\Xi}), \\label{p2ca}\\\\\n\t{\\mathrm{s.t.}} \\quad &{\\boldsymbol\\Xi} \\in \\mathcal{F}_3^{S\\times M}. \\label{p2cb}\n\t\\end{align}\n\\end{subequations}\nDue to the difficulty caused by the non-convexity of constraints, we start from the unconstrained problem as\n\\begin{align}\\label{eq:problem2cuncons}\n\\mathcal{P}_{2}^{ucst}:\\ \\underset{\\widetilde{\\bV}_1} \\max\\ \\log\\det\\left(\\mathbf{I}_{S}+\\widetilde{\\bV}_1^H\\mathbf{S}\\widetilde{\\bV}_1\\right),\n\\end{align}\nwhere $\\mathbf{S}=\\frac{1}{\\sigma^{2}}\\sum\\limits_{k=1}^{K}\\mathbf{H}_1\\bm{\\Phi}\\bH_{2,k}\\mathbf{Q}_k\\bH_{2,k}^H\\bm{\\Phi}^H\\mathbf{H}_1^H\\in\\mathbb{C}^{M\\times M}$, and $\\widetilde{\\bV}_1$ is denoted in \\subsecref{subsec:Qk1}. For problem $\\mathcal{P}_{2}^{ucst}$, \\cite[Corollary 2]{SDE19} indicates the closed form of the optimal solution. Denote the eigenvalue decomposition $\\mathbf{S}=\\mathbf{V}\\widetilde{\\mathbf{S}}\\mathbf{V}^H$ where the diagonal element of $\\widetilde{\\mathbf{S}}$ is arranged in the decending order, then $\\widetilde{\\mathbf{V}}_{1,\\mathrm{opt}}$ can achieve maximum sum rate of \\eqref{eq:problem2cuncons} when $\\widetilde{\\bV}_1$ is the first $S$ columns of $\\mathbf{V}$, i.e.,\n\\begin{align}\\label{equ:optimalV1}\n\\widetilde{\\mathbf{V}}_{1,\\mathrm{opt}}=\\mathbf{V}_{[1:S]}.\n\\end{align}\nSubstituting \\eqref{equ:optimalV1} into \\eqref{equ:Xidecompose}, we derive the optimal ${\\boldsymbol\\Xi}$ without the constraint. It is noteworthy that this result is independent of $\\mathbf{U}_1$ and $\\widetilde{\\bXi}$ in \\eqref{equ:Xidecompose}, thus arbitrary values of $\\mathbf{U}_1$ and $\\widetilde{\\bXi}$ exert no effect on the unconstrained maximal SE. With this in mind, we intend to approximate the optimal weight matrix ${\\boldsymbol\\Xi}$ satisfying the constraints by reasonably configuring these two values, which is formulated by\n\\begin{subequations}\\label{problem2ccons}\n\t\\begin{align}\n\t\\mathcal{P}_{2}^{cst}:\\ &\\underset{{\\boldsymbol\\Xi},\\mathbf{U}_1,\\widetilde{\\bXi}} \\min\\  \\big|\\big|{\\boldsymbol\\Xi}-\\mathbf{U}_1\\widetilde{\\bXi}\\widetilde{\\bV}_1^H\\big|\\big|_\\mathrm{F}^2, \\label{p2csta}\\\\\n\t{\\mathrm{s.t.}} \\quad &{\\boldsymbol\\Xi} \\in \\mathcal{F}_3^{S\\times M}, \\mathbf{U}_1\\in\\mathcal{U}^S, \\widetilde{\\bXi}\\in\\mathcal{D}^S, \\label{p2cstb}\n\t\\end{align}\n\\end{subequations}\nwhere $\\mathcal{U}^S$ and $\\mathcal{D}^S$ are the sets of unitary matrices and diagonal matrices, respectively, on the dimension of $S\\times S$. Resorting to the AO method, we address $\\mathcal{P}_{2}^{cst}$ by iteratively optimize three variables with the other two fixed as follows:\n\nFirst, as $\\mathbf{U}_1$ and $\\widetilde{\\bXi}$ are fixed, it is obvious that the optimal ${\\boldsymbol\\Xi}$ of \\eqref{problem2ccons} is dependent on $\\mathbf{T}=\\mathbf{U}_1\\widetilde{\\bXi}\\widetilde{\\bV}_1^H$, specifically,\n\\begin{align}\\label{equ:Xi2opt}\n&{\\boldsymbol\\Xi}_{s_1,(s_2-1)L+l}^{\\mathrm{opt}}=\\notag\\\\\n&\\left\\{\n\\begin{array}{cccc}\n\\argmin{\\xi_{s_1,l}\\in\\mathcal{F}_2}\\left|\\xi_{s_1,l}-\\mathbf{T}_{s_1,(s_2-1)L+l}\\right|^2  &s_1=s_2 \\\\\n0 &s_1\\neq s_2 \\\\\n\\end{array}\n\\right..\n\\end{align}\nSecond, denote $\\mathbf{T}_1=\\widetilde{\\bXi}\\widetilde{\\bV}_1^H$. As ${\\boldsymbol\\Xi}$ and $\\widetilde{\\bXi}$ are fixed, we assume the left and right singular vector matrices of ${\\boldsymbol\\Xi}\\mathbf{T}_1^H$ are $\\mathbf{U}_S$ and $\\mathbf{V}_S$, repectively. Then the optimal $\\mathbf{U}_1$ is the solution of Procrustes problem \\cite[Ch. 7.4.5]{MatAna}:\n\\begin{align}\\label{equ:U12opt}\n&\\mathbf{U}_{1,\\mathrm{opt}}=\\argmin{\\mathbf{U}_1\\in\\mathcal{U}^S}\\big|\\big|{\\boldsymbol\\Xi}-\\mathbf{U}_1\\mathbf{T}_1\\big|\\big|_\\mathrm{F}^2=\\mathbf{U}_S\\mathbf{V}_S^H.\n\\end{align}\nThird, as ${\\boldsymbol\\Xi}$ and $\\mathbf{U}_1$ are fixed, we denote $\\mathbf{T}_2={\\boldsymbol\\Xi}^H\\mathbf{U}_1=[\\mathbf{t}_{2,1},...\\mathbf{t}_{2,S}]$, $\\widetilde{\\bV}_1=[\\widetilde{\\mathbf{v}}_{1,1},...,\\widetilde{\\mathbf{v}}_{1,S}]$ for the equivalent problem\n\\begin{align}\\label{equ:Xitildeprob}\n&\\widetilde{\\bXi}_{\\mathrm{opt}}=\\argmin{\\widetilde{\\bXi}\\in\\mathcal{D}^S}\\big|\\big|\\mathbf{T}_2^H-\\widetilde{\\bXi}\\widetilde{\\bV}_1^H\\big|\\big|_\\mathrm{F}^2.\n\\end{align}\nThen, the diagonal entries of optimal $\\widetilde{\\bXi}$ can be written as\n\\begin{align}\\label{equ:Xitilde2opt}\n&\\widetilde{\\bXi}_{s,s}^{\\mathrm{opt}}=\\max\\left\\{\\frac{\\Re\\left\\{\\mathbf{t}_{2,s}^H\\widetilde{\\mathbf{v}}_{1,s}\\right\\}}{\\left|\\left|\\widetilde{\\mathbf{v}}_{1,s}\\right|\\right|_\\mathrm{F}^2},\\delta\\right\\},\\ \\forall s\\in\\{1,...,S\\},\n\\end{align}\nwhere $\\delta$ is an infinitesimal positive number \\cite[Lemma 2]{SDE19}. The DMA weight matrix optimization with $\\mathbf{Q}$ and $\\bm{\\Phi}$ fixed is detailed in \\textbf{\\alref{alg:AO_DMAs}}.\n\\begin{algorithm}[h]\n\t\\caption{DMA Weight Matrix Design with Full CSI}\n\t\\label{alg:AO_DMAs}\n\t\\begin{algorithmic}[1]\n\t\t\\State Initialize $\\mathbf{U}_1^{(0)},\\widetilde{\\bXi}^{(0)}$, iteration index $p=0$, thresholds $\\epsilon_3$.\n\t\t\\State Calculate $\\widetilde{\\mathbf{V}}_{1,\\mathrm{opt}}$ by \\eqref{equ:optimalV1} and then get ${\\boldsymbol\\Xi}^{(0)}$ by \\eqref{equ:Xi2opt}.\n\t\t\\Repeat\n\t\t\\State Obtain $\\mathbf{U}_{1}^{(p)}$ by \\eqref{equ:U12opt}.\n\t\t\\State Obtain the diagonal entries of $\\widetilde{\\bXi}^{(p)}$ by \\eqref{equ:Xitilde2opt}.\n\t\t\\State Set $p=p+1$.\n\t\t\\State Update the entries of ${\\boldsymbol\\Xi}^{(p)}$ by \\eqref{equ:Xi2opt}.\n\t\t\\Until $\\left|\\left|{\\boldsymbol\\Xi}^{(p)}-{\\boldsymbol\\Xi}^{(p-1)}\\right|\\right|_\\mathrm{F}^2\\leq \\epsilon_3$.\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\n\\section{EM Aware SE Optimization with Partial CSI} \\label{sec:opt_SE_design_SCSI}\nAs for the RIS-aided uplink, it is often the case that users are moving at high speed while RIS and BS are static in the system. Thus the rapid time-varying of the transmission from users to RIS is nonnegligible, which requires more accurate channel estimation and more frequent updates of the transmission design \\cite{WMJ13}. To this end, we assume that the RIS-to-DMAs channel $\\mathbf{H}_1$ is perfectly known while only statistical CSI of the users-to-RIS channels $\\{\\bH_{2,k}\\}_{\\forall k}$ is available.\n\nAs presented in \\cite{GJL09}, $\\bH_{2,k}$ can be decomposed with the adoption of Weichselberger's model as\n\\begin{align}\\label{equ:decomHtk}\n\\bH_{2,k}=\\mathbf{U}_{2,k}\\widetilde{\\bH}_{2,k}\\mathbf{V}_{2,k}^H\\in\\mathbb{C}^{N_R\\times N_k},\n\\end{align}\nwhere $\\mathbf{U}_{2,k}\\in\\mathbb{C}^{N_R\\times N_R}$ and $\\mathbf{V}_{2,k}\\in\\mathbb{C}^{N_k\\times N_k}$ are deterministic unitary matrices, and $\\widetilde{\\bH}_{2,k}\\in\\mathbb{C}^{N_R\\times N_k}$ is a random matrix with elements obeying zero-mean independent distribution. The statistics of CSI is presented by the eigenmode coupling matrix \\cite{WJW11}, i.e.,\n\\begin{align}\\label{equ:OmegaHtwok}\n\\bm{\\Omega}_{2,k}=\\mathbb{E}\\left\\{\\widetilde{\\bH}_{2,k}\\odot\\widetilde{\\bH}_{2,k}^H\\right\\}\\in \\mathbb{R}^{N_R\\times N_k}.\n\\end{align}\nIn addition, \\eqref{equ:SE_model} indicates the ergodic SE formulated by\n\\begin{align}\\label{equ:SE_model3}\n&{\\eta}_{\\mathrm{SE}}(\\mathbf{Q}, \\bm{\\Phi}, \\widetilde{\\bV}_1)=\\mathbb{E}_{\\{\\bH_{2,k}\\}}\\Bigg\\{\\log\\det\\Bigg(\\mathbf{I}_{S}+\\notag\\\\\n&\\qquad\\quad\\left.\\left.\\frac{1}{\\sigma^{2}}\\sum\\limits_{k=1}^{K}\\widetilde{\\bV}_1^H\\mathbf{H}_1\\bm{\\Phi}\\bH_{2,k}\\mathbf{Q}_k\\bH_{2,k}^H\\bm{\\Phi}^H\\mathbf{H}_1^H\\widetilde{\\bV}_1\\right)\\right\\}.\n\\end{align}\nNote that the existence of expectations introduces significant computational overhead. Especially for the Monte Carlo method, by which channel states are exhausted for the calculation of the average SE \\cite{LGX16}. A low complexity approach for addressing the expectation operations without averaging is the DE method. Via utilizing the large-dimensional random matrix theory, the DE method can provide deterministic approximations of functionals of random matrices, which are asymptotically accurate as the matrix dimensions grow to infinity at a fixed rate \\cite{CD11,WJW11}.\nIn this way, the accurate approximation of \\eqref{equ:SE_model3}, which is defined as ${\\overline {\\eta}}_{\\mathrm{SE}}$, can be obtained at a low computational level by using $\\bm{\\Omega}_{2,k},\\forall k$. Similar to \\secref{sec:opt_SE_design_PCSI}, we adopt AO method to iteratively optimize $\\mathbf{Q}$, $\\bm{\\Phi}$ and ${\\boldsymbol\\Xi}$.\n\n\\subsection{Transmit Covariance Matrices Design}\\label{subsec:Qk2}\n\n\\subsubsection{Deterministic Equivalent Method}\nAssume that $\\mathbf{G}_k=\\widetilde{\\bV}_1^H\\mathbf{H}_1\\bm{\\Phi}\\mathbf{U}_{2,k}\\widetilde{\\bH}_{2,k}\\mathbf{V}_{2,k}^H$, the ergodic SE in \\eqref{equ:SE_model3} is equivalent to \\eqref{p2aa} with the expectation of $\\{\\mathbf{G}_k\\}_{\\forall k}$, i.e.,\n\\begin{align}\\label{equ:Gkexp}\n\t{\\eta}_{\\mathrm{SE}}(\\mathbf{Q})=\\mathbb{E}_{\\{\\mathbf{G}_k\\}}\\log\\det\\left(\\mathbf{I}_{S}+\\frac{1}{\\sigma^{2}}\\sum\\limits_{k=1}^{K}\\mathbf{G}_k\\mathbf{Q}_k\\mathbf{G}_k^H\\right).\n\\end{align}\nDenote by $\\mathbf{U}_{\\mathrm{G}_k^a}=\\widetilde{\\bV}_1^H\\mathbf{H}_1\\bm{\\Phi}\\mathbf{U}_{2,k}\\in\\mathbb{C}^{S\\times N_R}$. Then, following closely to the proof of \\cite[Prop. 1]{WJW11}, the asymptotic approximation of \\eqref{equ:Gkexp} can be expressed by\n\\begin{align}\\label{equ:DEsGk}\n&{\\overline {\\eta}}_{\\mathrm{SE}}(\\mathbf{Q})=\\sum\\limits_{k=1}^{K} \\log\\det\\left(\\mathbf{I}_{N_k}+\\frac{1}{\\sigma^{2}}\\bm{\\Gamma}_{k}^a\\mathbf{Q}_k\\right)\n\\notag\\\\\n&\\quad\\quad+\\log\\det\\left(\\mathbf{I}_{S}+\\sum\\limits_{k=1}^{K}\\mathbf{\\Psi}_{k}^a\\right)-\\sum\\limits_{k=1}^{K}(\\bm{\\gamma}_{k}^a)^T\\bm{\\Omega}_{2,k}\\bm{\\psi}_{k}^a,\n\\end{align}\nwhere $\\bm{\\gamma}_{k}^a \\triangleq [\\gamma_{k,1}^{a},...\\gamma_{k,N_R}^{a}]^T$, $\\bm{\\psi}_{k}^a \\triangleq [\\psi_{k,1}^{a},...\\psi_{k,N_k}^{a}]^T$ are DE parameters. The other two parameters $\\bm{\\Gamma}_{k}^a$ and $\\mathbf{\\Psi}_{k}^a$ are calculated by\n\\begin{align}\n\\label{equ:DE_para_Gammak_Gk}\n\\bm{\\Gamma}_{k}^a&=\\mathbf{V}_{2,k}\\mathrm{diag}\\{\\bm{\\Omega}_{2,k}^T\\bm{\\gamma}_{k}^a\\}\\mathbf{V}_{2,k}^H \\in \\mathbb{C}^{N_k \\times N_k},\\\\\n\\label{equ:DE_para_Psik_Gk}\n\\mathbf{\\Psi}_{k}^a&=\\mathbf{U}_{\\mathrm{G}_k^a}\\mathrm{diag}\\{\\bm{\\Omega}_{2,k}\\bm{\\psi}_{k}^a\\}\\mathbf{U}_{\\mathrm{G}_k^a}^H \\in \\mathbb{C}^{S \\times S}.\n\\end{align}\nAssume that $r\\in\\{1,...,N_R\\}$ and $n_k\\in\\{1,...,N_k\\}$, $\\mathbf{u}_{\\mathrm{G}_{k}^a,r}$ and $\\mathbf{v}_{2,n_k}$ represent the $r$th column and $n_k$th column vectors of $\\mathbf{U}_{\\mathrm{G}_k^a}$ and $\\mathbf{V}_{2,k}$, respectively, then\n\\begin{align}\n\\label{equ:DE_para_gammak_Gk}\n\\gamma_{k,r}^{a}&=\\mathbf{u}_{\\mathrm{G}_{k}^a,r}^H(\\mathbf{I}_{S}+\\sum\\limits_{k=1}^{K}\\mathbf{\\Psi}_{k}^a)^{-1}\\mathbf{u}_{\\mathrm{G}_{k}^a,r},\\ \\forall k,r,\\\\\n\\label{equ:DE_para_psik_Gk}\n\\psi_{k,n_k}^{a}&=\\mathbf{v}_{2,n_k}^H\\mathbf{Q}_k(\\sigma^{2}\\mathbf{I}_{N_k}+\\bm{\\Gamma}_{k}^a\\mathbf{Q}_k)^{-1}\\mathbf{v}_{2,n_k},\\ \\forall k,n_k.\n\\end{align}\nWith an initial point of $\\bm{\\psi}_{k}^a$, the DE parameters $\\bm{\\Gamma}_{k}^a$ and $\\mathbf{\\Psi}_{k}^a$ that are used to approximate the ergodic SE can be obtained by cyclically updating $\\bm{\\gamma}_{k}^a$ and $\\bm{\\psi}_{k}^a$ through \\eqref{equ:DE_para_Gammak_Gk} -- \\eqref{equ:DE_para_psik_Gk}, which inspires us to propose an iterative algorithm as shown in \\textbf{\\alref{alg:DEs}}.\n\\begin{algorithm}[h]\n\t\\caption{Deterministic Equivalent Algorithm}\n\t\\label{alg:DEs}\n\t\\begin{algorithmic}[1]\n\t\t\\State Initialize $\\{(\\bm{\\psi}_{k}^{a})^{(0)}\\}$, iteration index $q=0$, threshold $\\epsilon_4$.\n\t\t\\For{$k=1$ to $K$}\n\t\t\\Repeat\n\t\t\\State Calculate $(\\bm{\\gamma}_{k}^{a})^{(q+1)}$ by \\eqref{equ:DE_para_Psik_Gk} and \\eqref{equ:DE_para_gammak_Gk}.\n\t\t\\State Update $(\\bm{\\psi}_{k}^{a})^{(q+1)}$ by \\eqref{equ:DE_para_Gammak_Gk} and \\eqref{equ:DE_para_psik_Gk} with $\\{\\mathbf{Q}_k\\}_{\\forall k}$.\n\t\t\\State Set $q=q+1$.\n\t\t\\Until $\\left|\\left|(\\bm{\\psi}_{k}^{a})^{(q)}-(\\bm{\\psi}_{k}^{a})^{(q-1)}\\right|\\right|^2\\leq \\epsilon_4$.\n\t\t\\State Calculate $\\bm{\\Gamma}_{k}^a$ and $\\mathbf{\\Psi}_{k}^a$ by \\eqref{equ:DE_para_Gammak_Gk} and \\eqref{equ:DE_para_Psik_Gk} with $(\\bm{\\gamma}_{k}^{a})^{(q)}$ and $(\\bm{\\psi}_{k}^{a})^{(q)}$.\n\t\t\\EndFor\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\\subsubsection{EM-Aware Modified Water-filling Algorithm}\nNote that ${\\overline {\\eta}}_{\\mathrm{SE}}$ in \\eqref{equ:DEsGk} is a tight approximation even when $N_k$ and $N_R$ are small \\cite{WJW11}. Replacing \\eqref{p2aa} by \\eqref{equ:DEsGk}, we arrive at the asymptotic SE optimization problem as follows:\n\\begin{subequations}\\label{eq:problem3a}\n\t\\begin{align}\n\t\\mathcal{P}_3^a:\\quad\\underset{\\{\\mathbf{Q}_k\\}} \\max \\quad &{\\overline {\\eta}}_{\\mathrm{SE}}(\\mathbf{Q}), \\label{p3a}\\\\\n\t{\\mathrm{s.t.}} \\quad &\\tr{\\mathbf{Q}_k}\\leqP_{\\mathrm{max},k},\\quad \\mathbf{Q}_k \\succeq \\mathbf{0}, \\label{p3b}\\\\\n\t&\\tr{\\mathbf{R}_{k,i}\\mathbf{Q}_k}\\leq D_{k,i},\\quad \\forall k,i. \\label{p3c}\n\t\\end{align}\n\\end{subequations}\nSimilar to \\emph{\\propref{Prop:Qk_optimal1}}, the optimal $\\mathbf{Q}_k$ with partial CSI is also available by a modified water-filling algorithm.\n\\begin{prop}\\label{Prop:Qk_optimal2}\n\tAssume $\\{\\mu_{k,\\mathrm{opt}}\\}_{\\forall k}$ and $\\{\\lambda_{k,i,\\mathrm{opt}}\\}_{\\forall k,i}$ are optimal dual values of power and SAR constraints, respectively, and $\\mathbf{K}_{k,\\mathrm{opt}}=\\mu_{k,\\mathrm{opt}}\\mathbf{I}_{N_k}+\\sum\\limits_{i=1}^{A_k}\\lambda_{k,i,\\mathrm{opt}}\\mathbf{R}_{k,i}$. Denote the eigenvalue decomposition \n\t\\begin{align}\\label{Kk_Gammak_eig_decomp}\n\t\\mathbf{K}_{k,\\mathrm{opt}}^{-1\/2}\\bm{\\Gamma}_{k}^a\\mathbf{K}_{k,\\mathrm{opt}}^{-1\/2}=\\mathbf{U}_k{\\boldsymbol\\Sigma}_k\\mathbf{U}_k^H.\t\n\t\\end{align}\n\tThen, $\\mathbf{Q}_{k,\\mathrm{opt}}$ can be obtained by the same method as \\eqref{Optimal_Q1} and \\eqref{equ:Lambdak} in \\emph{\\propref{Prop:Qk_optimal1}}.\n\\end{prop}\n\\IEEEproof Refer to Appendix B.\n\nPlease note that $\\bm{\\Gamma}_{k}^a$ in \\eqref{Kk_Gammak_eig_decomp} also depends on $\\{\\mathbf{Q}_{k,\\mathrm{opt}}\\}_{\\forall k}$. We also applies the AO method, i.e., cyclically calculating $\\mathbf{Q}_{k,\\mathrm{opt}}$ in \\emph{\\propref{Prop:Qk_optimal2}} with given $\\bm{\\Gamma}_{k}^a$, then update $\\bm{\\Gamma}_{k}^a$ through the newly adjusted $\\mathbf{Q}_{k,\\mathrm{opt}}$, which is reflected in steps 4-10 of \\textbf{\\alref{alg:waterfilling_SCSI}}.\n\\begin{algorithm}[h]\n\t\\caption{Optimization of $\\{\\mathbf{Q}_{k}\\}_{\\forall k}$ with Partial CSI}\n\t\\label{alg:waterfilling_SCSI}\n\t\\begin{algorithmic}[1]\n\t\t\\State Initialize dual variables $\\mu_{k}^{(0)}$, $\\lambda_{k,i}^{(0)},\\forall k,i$, feasible $\\mathbf{Q}_k^{(0)},\\forall k$, iteration indices $u_1=0$, $u_2$, threshold $\\epsilon_5$. \n\t\t\\Repeat\n\t\t\\State Set $\\{\\mathbf{Q}_{k,(0)}^{(u_1)}\\}_{\\forall k}=\\{\\mathbf{Q}_k^{(u_1)}\\}_{\\forall k}$, $u_2=0$.\n\t\t\\Repeat\n\t\t\\State Calculate DE parameters $\\bm{\\Gamma}_{k,(u_2)}^{a}$ and $\\mathbf{\\Psi}_{k,(u_2)}^{a}, \\forall k$ by $\\textbf{\\alref{alg:DEs}}$ with given $\\mathbf{Q}_{k,(u_2)}^{(u_1)},{\\forall k}$.\n\t\t\\State Obtain $\\mathbf{U}_{k,(u_2)}$ and ${\\boldsymbol\\Lambda}_{k,(u_2)},\\forall k$ by \\eqref{Kk_Gammak_eig_decomp} and \\eqref{equ:Lambdak}.\n\t\t\\State Calculate $\\mathbf{Q}_{k,(u_2+1)}^{(u_1)},{\\forall k}$ by \\eqref{Optimal_Q1}.\n\t\t\\State Set $u_2=u_2+1$.\n\t\t\\Until $\\left| {\\overline {\\eta}}_{\\mathrm{SE}}\\left(\\mathbf{Q}_{k,(u_2)}^{(u_1)}\\right)-{\\overline {\\eta}}_{\\mathrm{SE}}\\left(\\mathbf{Q}_{k,(u_2-1)}^{(u_1)}\\right) \\right| \\leq \\epsilon_5$.\n\t\t\\State Return $\\{\\mathbf{Q}_k^{(u_1+1)}\\}_{\\forall k}=\\{\\mathbf{Q}_{k,(u_2)}^{(u_1)}\\}_{\\forall k}$.\n\t\t\\State Set $u_1=u_1+1$.\n\t\t\\State Update $\\mu_{k}^{(u_1)}$, $\\lambda_{k,i}^{(u_1)},\\forall k,i$ by minimizing $\\mathcal{L}$ in \\eqref{equ:dual_problem1}.\n\t\t\\Until $\\{\\mu_k\\}_{\\forall k}$, $\\{\\lambda_{k,i}\\}_{\\forall k,i}$ converge.\n\t\\end{algorithmic}\n\\end{algorithm}  \n\n\n\\subsection{RIS Phase Shift Matrix Design}\\label{subsec:bPhi2}\nWith fixed $\\mathbf{Q}$ and ${\\boldsymbol\\Xi}$, the asymptotic approximation of SE, represented by ${\\overline {\\eta}}_{\\mathrm{SE}}(\\bm{\\Phi})$, has the same form as \\eqref{equ:DEsGk}. In the DE method, we obtain the DE parameter $\\bm{\\Gamma}_{k}^a$ and $\\mathbf{\\Psi}_{k}^a$ with fixed $\\bm{\\gamma}_{k}^a$ and $\\bm{\\psi}_{k}^a$, thus only the second term of \\eqref{equ:DEsGk} is the function of $\\bm{\\Phi}$ while the other terms are considered as constants in the optimization of $\\bm{\\Phi}$ \\cite{YXN21}. Substituting \\eqref{equ:DE_para_Psik_Gk} into \\eqref{equ:DEsGk}, we express the problem as\n\\begin{subequations}\\label{eq:problem3b}\n\t\\begin{align}\n\t&\\mathcal{P}_{3}^{b}:\\ \\underset{\\bm{\\Phi}} \\max\\ \\log\\det\\Bigg(\\mathbf{I}_{S}+\\notag\\\\\n\t&\\  \\sum\\limits_{k=1}^{K}\\widetilde{\\bV}_1^H\\mathbf{H}_1\\bm{\\Phi}\\mathbf{U}_{2,k}\\mathrm{diag}\\{\\bm{\\Omega}_{2,k}\\bm{\\psi}_{k}^a\\}\\mathbf{U}_{2,k}^H\\bm{\\Phi}^H\\mathbf{H}_1^H\\widetilde{\\bV}_1\\Bigg), \\label{p3ba}\\\\\n\t&\\qquad\\quad{\\mathrm{s.t.}} \\ \\phi_n \\in \\mathcal{F}_1,\\quad \\forall n. \\label{p3bb}\n\t\\end{align}\n\\end{subequations}\nWe denote by \n\\begin{align}\\label{equ:Ptilde}\n\t\\widetilde{\\mathbf{P}}=\\sigma^{2}\\sum\\limits_{k=1}^{K}\\mathbf{U}_{2,k}\\mathrm{diag}\\{\\bm{\\Omega}_{2,k}\\bm{\\psi}_{k}^a\\}\\mathbf{U}_{2,k}^H,\n\\end{align}\nthen problem $\\mathcal{P}_{3}^{b}$ is equivalent to $\\mathcal{P}_{2}^{b}$ that just replaces $\\mathbf{P}$ by $\\widetilde{\\mathbf{P}}$, i.e.,\n\\begin{subequations}\\label{eq:problem3bplus}\n\t\\begin{align}\n\t\\mathcal{P}_{3}^{b}:\\underset{\\bm{\\Phi}} \\max\\ &\\log\\det\\left(\\mathbf{I}_{S}+\\frac{1}{\\sigma^{2}}\\widetilde{\\bV}_1^H\\mathbf{H}_1\\bm{\\Phi}\\widetilde{\\mathbf{P}}\\bm{\\Phi}^H\\mathbf{H}_1^H\\widetilde{\\bV}_1\\right), \\label{p3baa}\\\\\n\t{\\mathrm{s.t.}} \\ &\\phi_n \\in \\mathcal{F}_1,\\quad \\forall n. \\label{p3bbb}\n\t\\end{align}\n\\end{subequations}\nTherefore, the optimal $\\bm{\\Phi}$ with partial CSI will be derived by the same method in \\subsecref{subsec:bPhi1}. Above we consider the case where $\\bm{\\psi}_{k}^a$ is fixed. However, as $\\mathbf{U}_{\\mathrm{G}_k^a}$ is one of the matrices for calculating DE parameters, the value of $\\bm{\\psi}_{k}^a$ is required to be updated along with the variation of $\\bm{\\Phi}$.\n\n\\subsection{DMA Weight Matrix Design}\nSimilar to \\subsecref{subsec:bXi1}, we firstly consider the unconstrained problem and utilize the decomposition of \\eqref{equ:Xidecompose}. Based on the DE method, the asymptotic approximation of the system SE ${\\overline {\\eta}}_{\\mathrm{SE}}(\\widetilde{\\bV}_1)$ can also be expressed by \\eqref{equ:DEsGk}, where only the second term that contains $\\mathbf{U}_{\\mathrm{G}_k^a}$ is related to $\\widetilde{\\bV}_1$ while the others are constants. On this condition, we obtain the same objective function as \\eqref{p3ba}. Note that when denoting\n\\begin{align}\\label{equ:Stilde}\n\t\\widetilde{\\mathbf{S}}=\\mathbf{H}_1\\bm{\\Phi}\\mathbf{U}_{2,k}\\mathrm{diag}\\{\\bm{\\Omega}_{2,k}\\bm{\\psi}_{k}^a\\}\\mathbf{U}_{2,k}^H\\bm{\\Phi}^H\\mathbf{H}_1^H,\n\\end{align}\nthe optimization problem of ${\\boldsymbol\\Xi}$ without the constraint is expressed by \\begin{align}\\label{eq:problem3cuncons}\n\\mathcal{P}_{3}^{ucst}:\\ \\underset{\\widetilde{\\bV}_1} \\max\\ \\log\\det\\left(\\mathbf{I}_{S}+\\widetilde{\\bV}_1^H\\widetilde{\\mathbf{S}}\\widetilde{\\bV}_1\\right),\n\\end{align}\nwhich is converted to the same problem form as \\eqref{eq:problem2cuncons}. In the following, the optimization process of constrained ${\\boldsymbol\\Xi}$ is similar to the method in \\subsecref{subsec:bXi1}, thus omitted here. Note that when $\\bm{\\psi}_{k}^a$ is considered unfixed, the value of $\\widetilde{\\mathbf{S}}$ is related to $\\widetilde{\\bV}_1$, which inspires us to update the value of $\\widetilde{\\mathbf{S}}$ in the overall AO method.\n\n\\section{Overall Algorithm and Analysis}\\label{sec:analysis}\nNow that we have studied the optimization of covariance matrices $\\{\\mathbf{Q}\\}_{\\forall k}$, RIS phase shift matrix $\\bm{\\Phi}$ and DMA weight matrix ${\\boldsymbol\\Xi}$ separately under both full and partial CSI. In this section, we propose the overall EM aware SE maximization algorithm and then analyze its convergence and complexity. Note that whether with full or partial CSI, the overall algorithm shares the same AO-based framework. Therefore, we focus our analysis on the case with partial CSI, which has been studied in \\secref{sec:opt_SE_design_SCSI}. Then the convergence performance and the complexity with full CSI can be obtained in a similar way. \n\nCompared with \\secref{sec:opt_SE_design_PCSI} that consider the full CSI case, \\secref{sec:opt_SE_design_SCSI} adopts the DE method to approximate the system SE without requiring time-consuming expectation operation. The DE parameters used for asymptotic approximation can be obtained by using iteration in \\textbf{\\alref{alg:DEs}} until $\\bm{\\gamma}_{k}^a$ and $\\bm{\\psi}_{k}^a$ converge to the unique solution point \\cite{WJW11}. By using the AO-based framework, the overall transmission strategy with partial CSI is detailed in \\textbf{\\alref{alg:AOmethod}}. \nSimilarly, we get the AO-based algorithm with full CSI i.e., iterating \\textbf{Algorithm 1, 2, 3} until the system performance converges.\n\n\\begin{algorithm}[h]\n\t\\caption{AO-based Method for SE Maximization with Partial CSI}\n\t\\label{alg:AOmethod}\n\t\\begin{algorithmic}[1]\n\t\t\\Require Initial $\\mathbf{Q}^{(0)}$, $\\bm{\\Phi}^{(0)}$, ${\\boldsymbol\\Xi}^{(0)}$, iteration index $\\ell$, threshold $\\epsilon_6$.\n\t\t\\Ensure Approximation solutions $\\mathbf{Q}_{\\mathrm{opt}}$, $\\bm{\\Phi}_{\\mathrm{opt}}$, and ${\\boldsymbol\\Xi}_{\\mathrm{opt}}$.\n\t\t\\State Set $\\ell=0$ and calculate ${\\overline {\\eta}}_{\\mathrm{SE}}^{(0)}$ by \\eqref{equ:DEsGk}.\n\t\t\\Repeat \n\t\t\\State Obtain $\\mathbf{Q}^{(\\ell+1)}=\\mathrm{diag}\\{\\mathbf{Q}_k^{(\\ell+1)}\\}_{k=1}^{K}$ by \\textbf{\\alref{alg:waterfilling_SCSI}}.\n\t\t\\State Get $\\mathbf{m}{\\psi}_{k}^a$ by \\textbf{\\alref{alg:DEs}} and calculate $\\widetilde{\\mathbf{P}}^{(\\ell)}$ by \\eqref{equ:Ptilde}.\n\t\t\\State Obtain $\\bm{\\Phi}^{(\\ell+1)}$ by \\textbf{\\alref{alg:WMMSE_PCSI}}.\n\t\t\\State Get $\\mathbf{m}{\\psi}_{k}^a$ by \\textbf{\\alref{alg:DEs}} and calculate $\\widetilde{\\mathbf{S}}^{(\\ell)}$ by \\eqref{equ:Stilde}.\n\t\t\\State Obtain ${\\boldsymbol\\Xi}^{(\\ell+1)}$ by \\textbf{\\alref{alg:AO_DMAs}}.\n\t\t\\State Update $\\ell=\\ell+1$.\n\t\t\\State Calculate ${\\eta}_{\\mathrm{SE}}^{(\\ell)}$ with $\\mathbf{Q}^{(\\ell)}$, $\\bm{\\Phi}^{(\\ell)}$ and ${\\boldsymbol\\Xi}^{(\\ell)}$ by \\eqref{equ:DEsGk}.\n\t\t\\Until $\\left|{\\overline {\\eta}}_{\\mathrm{SE}}^{(\\ell)}-{\\overline {\\eta}}_{\\mathrm{SE}}^{(\\ell-1)}\\right|\\leq \\epsilon_6$.\n\t\t\\State Return $\\mathbf{Q}_{\\mathrm{opt}}=\\mathbf{Q}^{(\\ell)}$, $\\bm{\\Phi}_{\\mathrm{opt}}=\\bm{\\Phi}^{(\\ell)}$ and ${\\boldsymbol\\Xi}_{\\mathrm{opt}}={\\boldsymbol\\Xi}^{(\\ell)}$.\n\t\\end{algorithmic}\n\\end{algorithm}\n\nIn general, \\textbf{\\alref{alg:AOmethod}} adopts the method that $\\mathbf{Q}$, $\\bm{\\Phi}$ and ${\\boldsymbol\\Xi}$ are optimized alternatively. Due to the fact that \\eqref{eq:problem2a} and \\eqref{eq:problem3a} are convex problems, the solutions of \\emph{\\propref{Prop:Qk_optimal1}} and \\emph{\\propref{Prop:Qk_optimal2}} are essentially the results of solving their strong dual problems like \\eqref{equ:dual_problem1}. Therefore, the water-filling algorithm by iterating dual variables in \\textbf{\\alref{alg:waterfilling_PCSI}} and \\textbf{\\alref{alg:waterfilling_SCSI}} will definitely converges to the global optimal solutions of problems \\eqref{eq:problem2a} and \\eqref{eq:problem3a}, respectively \\cite{Convex}. In addition, \\textbf{\\alref{alg:WMMSE_PCSI}} solve the WMMSE problem in \\eqref{eq:problem2bMMSE} by adopting BCD method, of which the convergence is ensured according to \\cite{SRL11}. \\textbf{\\alref{alg:AO_DMAs}} utilizes the AO method for problem \\eqref{problem2ccons} with multiple optimization variables. As the object function in \\eqref{p2csta} is differentiable over all the variables, the AO algorithm is ensured to converge \\cite{BH03}. Therefore, the value of system SE will not reduce through steps 3, 5, 7 in \\textbf{\\alref{alg:AOmethod}}, which guarantees the convergence of the overall algorithm. Likewise, the convergence under full CSI scenario can be ensured by the fact the optimization objective is upper-bounded and does not decrease after each AO process.\n\nNotice that the complexity of the proposed algorithm is related to the number of iterations, which can be adjusted by varying the input thresholds. As the result of the fast convergence of DE parameters, the complexity of \\textbf{\\alref{alg:DEs}} can be almost ignored [47]. In \\textbf{\\alref{alg:waterfilling_SCSI}}, the major complexity of steps 5--8 locates in (50) and can be expressed by $\\mathcal{O}(N_k^3)$. Moreover, step 12 updates dual variables through minimizing $\\mathcal{L}$, which has the complexity of $\\mathcal{O}((K+\\sum_{k}A_k)^x)$, where $1\\leq x\\leq 4$ for the convex program [33]. Suppose the numbers of iterations for the inner AO and the outer water-filling are $I_{\\mathrm{A}}$ and $I_{\\mathrm{W}}$, respectively. Then, the complexity of \\textbf{\\alref{alg:waterfilling_SCSI}} is estimated with $\\mathcal{O}(I_{\\mathrm{W}}(I_{\\mathrm{A}}\\sum_{k}N_k^3+(K+\\sum_{k}A_k)^x))$. As \\textbf{\\alref{alg:waterfilling_PCSI}} has no need to obtain the optimal DE parameters $\\bm{\\Gamma}_{k}^a$, its complexity can be witten as $\\mathcal{O}(I_{\\mathrm{W}}(\\sum_{k}N_k^3+(K+\\sum_{k}A_k)^x))$. \\textbf{\\alref{alg:WMMSE_PCSI}} is composed of the outer BCD and the inner MM algorithms, which are assumed to go through $I_{\\mathrm{B}}$ and $I_{\\mathrm{M}}$ iterations, respectively. Because of the matrix inversion operations in (23) and (24), their complexity are $\\mathcal{O}(N_R^3)$ and $\\mathcal{O}(S^3)$, respectively. In step 6, the eigenvalues of $\\mathbf{m}{\\Delta}$ need to be pre-calculated, which introduces the complexity of $\\mathcal{O}(N_R^3)$. In addition, the complexity of MM algorithm mainly centers at the calculation of $\\mathbf{c}^{(\\zeta)}$, which is $\\mathcal{O}(N_R^2)$. Considering that usually $N_R^3\\gg S^3$, the complexity of \\textbf{\\alref{alg:WMMSE_PCSI}} can be estimated as $\\mathcal{O}(I_{\\mathrm{B}}(N_R^3+I_{\\mathrm{M}}N_R^2))$. The calculation of each iteration in \\textbf{\\alref{alg:AO_DMAs}} is concentrated in (37), which has the complexity of $\\mathcal{O}(S^3)$. Assume the number of iterations of \\textbf{\\alref{alg:AO_DMAs}} is $I_{\\mathrm{O}}$, its complexity is estimated with $\\mathcal{O}(I_{\\mathrm{O}}S^3)$. Based on the analysis above, the complexity of \\textbf{\\alref{alg:AOmethod}} is expressed as $\\mathcal{O}(I_{\\mathrm{AO}}(I_{\\mathrm{W}}(I_{\\mathrm{A}}\\sum_{k}N_k^3+(K+\\sum_{k}A_k)^x)+I_{\\mathrm{B}}(N_R^3+I_{\\mathrm{M}}N_R^2)+I_{\\mathrm{O}}S^3)$, where $I_{\\mathrm{AO}}$ is the number of iterations of the overall AO. Adopting the same notations, the AO-based algorithm with full CSI has the above complexity form removing the number of iterations, $I_{\\mathrm{A}}$.\n\n\n\\section{Numerical Results}\\label{sec:numerical_results}\nIn this section, the simulation results are given to appraise the SE performance of the EM aware transmission strategy in which RIS and DMAs are considered in the multiuser MIMO system. Our simulation focuses on the main contributions of the proposed algorithm, i.e, the SE maximization method under both full CSI and partial CSI. In practical scenarios, $\\mathcal{F}_3^{S\\times M}$ in \\eqref{p1e} is determined by the ability to externally control the response of each element, which comprises of four classical DMA weights \\cite{SDE19,WSE21}, i.e.,\n\\begin{itemize}\n\t\\item[(1).] $\\mathrm{{UC}}$: $\\mathcal{F}_2=\\mathbb{C}$.\n\t\\item[(2).] $\\mathrm{{AO}}$: $\\mathcal{F}_2=[0.001,2]$.\n\t\\item[(3).] $\\mathrm{{BA}}$: $\\mathcal{F}_2=\\{0,0.1\\}$.\n\t\\item[(4).] $\\mathrm{{LP}}$:  $\\mathcal{F}_2=\\left\\{\\frac{\\jmath+\\mathrm{e}^{\\jmath\\varphi}}{2}|\\varphi\\in\\left[\\right.0,2\\pi\\left.\\right)\\right\\}$.\n\\end{itemize}\nNotice that our simulations are mainly based on unconstrained DMA weights, i.e., $\\mathcal{F}_2=\\mathbb{C}$, unless explicitly mentioned otherwise. The parameters used in the simulation are listed in \\tabref{parameters} \\cite{YLH17,XYN21}.\n\\begin{table}[htbp]\n\t\\caption{Simulation Parameters}\\label{parameters}\n\t\\centering\n\t\\ra{1.3}\n\t\\footnotesize\n\t\\begin{tabular}{LR}\n\t\t\\toprule\n\t\tParameter & Value  \\\\\n\t\t\\midrule\n\t\t\\rowcolor{lightblue}\n\t\tChannel model     & 3GPP SCM    \\\\\n\t\tValues of $\\mathbf{U}_{2,k}$ and $\\mathbf{V}_{2,k},\\forall k$ & DFT matrices\\\\\n\t\t\\rowcolor{lightblue}\n\t\tNoise covariance                & -96 dBm  \\\\\n\t\tPath loss      &120 dB                            \\\\\n\t\t\\rowcolor{lightblue}\n\t\tNumber of users &$K=4$\\\\\n\t\tNumber of attennas at users & $N_k=4,\\forall k$\\\\\n\t\t\\rowcolor{lightblue}\n\t\tNumber of SAR constraints & $A_k=1,\\forall k$\\\\\n\t\tNumber of RIS phase shift units & $N_R=16$ \\\\\n\t\t\\rowcolor{lightblue}\n\t\tNumber of DMA microstrips & $S=8$\\\\\n\t\tNumber of metamaterial units per microstrip & $L=8$\\\\\n\t\t\\bottomrule\n\t\\end{tabular}\n\\end{table}\n\nWithout loss of generality, the power constraints and SAR constraints are set to be the same for all users for the clarity of the simulation results \\cite{YLH17}. Therefore, we assume that $P_{\\mathrm{max},k}=P_{\\mathrm{max}}, \\forall k$ and $D_{k,i}=D=0.8\\ \\mathrm{W\/kg}, \\forall k,i$ in our simulation. The SAR matrix corresponding to the SAR budget $D$ is assigned as \\cite{YLH15}\n\\begin{align}\n\\mathbf{R}_{k,i}=\\mathbf{R}=\n\\left[\n\\begin{array}{cccc}\n8 &-6\\jmath &-2.1 &0 \\\\\n6\\jmath &8 &-6\\jmath &-2.1 \\\\\n-2.1 &6\\jmath &8 &-6\\jmath \\\\\n0 &-2.1 &6\\jmath &8\n\\end{array}\n\\right].\n\\end{align}\n\n\\subsection{Convergence Performance}\n\\begin{figure}[t]\n\t\\centering\n\t\\subfloat[]{\\centering\\includegraphics[width=0.35\\textwidth]{iterICSI.eps}\\label{fig:iterICSI}}\n\t\\hfill\n\t\\subfloat[]{\\centering\\includegraphics[width=0.35\\textwidth]{iterSCSI.eps}\\label{fig:iterSCSI}}\n\t\\caption{Convergence of the AO-based method for EM exposure constrained SE maximization design in hybrid RIS and DMA assisted system. (a) Full CSI;  (b) Partial CSI.}\n\t\\label{fig:iteration}\n\\end{figure}\n\n\\figref{fig:iteration} evaluates the number of iterations required for the proposed EM aware SE maximization method in RIS and DMA assisted system, where we jointly optimize the covariance matrix $\\mathbf{Q}$, RIS phase shift matrix $\\bm{\\Phi}$ and DMA weight matrix ${\\boldsymbol\\Xi}$ to maximize system SE with the determined values of SAR matrix $\\mathbf{R}$ and SAR budget $D$. Note that \\subfigref{fig:iteration}{fig:iterSCSI} presents the iteration state in \\textbf{\\alref{alg:AOmethod}} and \\subfigref{fig:iteration}{fig:iterICSI} corresponds to the overall algorithm with full CSI. \nThe results illustrate the rapid convergence rates of the partial CSI scenario in specific power budget regions compared with the full CSI scenario. However, when the perfect CSI of $\\bH_{2,k}$ is available, the computational complexity of the overall algorithm is greatly reduced, which leads to the fast running speed of the overall algorithm.\n\n\\subsection{Impact of EM Exposure and RIS on SE Optimization}\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.35\\textwidth]{sepDifSAR.eps}\n\t\\caption{Comparison of SE performance between the optimization with and without EM exposure constraints.}\n\t\\label{fig:sepnoSAR}\n\\end{figure}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.35\\textwidth]{sepUCNoRIS.eps}\n\t\\caption{Comparison of SE performance between the RIS assisted system and the non-RIS assisted system.}\n\t\\label{fig:sepUCnoRIS}\n\\end{figure}\n\n\\figref{fig:sepnoSAR} sketches the trend of overall SE performance versus the variation of power budget $P_{\\mathrm{max}}$ in the presence and absence of EM exposure constraints, respectively. It can be observed that the system SE in our work is not positively correlated with the transmit power as traditional ways, but first increases speedily with the increase of $P_{\\mathrm{max}}$, then become constant when $P_{\\mathrm{max}}$ is relatively large. This is mainly due to the EM exposure constraints introduced into the problem, i.e., \\eqref{p1c}. When there is a tiny power budget, $P_{\\mathrm{max}}$ is the main factor limiting the growth of system SE where SAR constraints are easily satisfied even all the power is used. With the increase of $P_{\\mathrm{max}}$, SAR constraints gradually become the main restriction of the system SE and eventually lead to the saturation of the curves. Actually, this process is affected by the SAR budget $D$. As shown in \\figref{fig:sepnoSAR}, the larger SAR budget means more relaxed SAR constraints, which brings higher SE performance and requires more power budget to achieve the saturation of the curves.\n\n\\figref{fig:sepUCnoRIS} compares the SE performance of the proposed approach in the RIS assisted system with the non-RIS assisted system when DMAs are configured at the BS. As expected, for both full CSI and partial CSI, optimization without RIS results in the reduction on the system SE compared to that with optimized RIS shift matrix $\\bm{\\Phi}$. In fact, if the $\\bm{\\Phi}_{\\mathrm{opt}}$ optimized by \\textbf{\\alref{alg:AOmethod}} is an identity matrix, the system with or without RIS assist will achieve the same SE. In addition, the optimization of the RIS phase shift matrix is related to the optimal transmission covariance matrix, as shown in steps 4, 5 of \\textbf{\\alref{alg:AOmethod}}. Therefore, the solution $\\bm{\\Phi}_{\\mathrm{opt}}$ under different power budgets will not be unified and require independent optimization, which emphasizes the necessity of joint optimization design.\n\n\\subsection{Comparison With Conventional Antennas}\n\\figref{fig:sepSCSIconventional} compares the SE performance of two scenarios when the BS is equipped with DMAs and that with conventional antennas. In the conventional system, the BS uses uniform linear arrays (ULAs) with antenna spacing of half-wavelength, where each antenna is associated with an independent RF chain \\cite{YXN21}. \nSuppose the same condition that the BS is placed with $M=SL=64$ antennas, which constitute a total of $64$ RF chains, and then the system SE is denoted by\n\\begin{align}\\label{equ:SE_conventional}\n&{\\eta}_{\\mathrm{SE}}=\\mathbb{E}_{\\{\\bH_{2,k}\\}}\\Bigg\\{\\log\\det\\Bigg(\\mathbf{I}_{M}+\\frac{1}{\\sigma^{2}}\\sum\\limits_{k=1}^{K}\\mathbf{H}_1\\cdot\\notag\\\\\n&\\qquad\\qquad\\bm{\\Phi}\\bH_{2,k}\\mathbf{Q}_k\\bH_{2,k}^H\\bm{\\Phi}^H\\mathbf{H}_1^H\\Bigg)\\Bigg\\}\\ \\mathrm {bps}\/\\mathrm {Hz}.\n\\end{align}\nIt is evident in \\figref{fig:sepSCSIconventional} that due to more RF chains at the BS side, the conventional antenna assisted system achieves higher SE compared with the DMA assisted system. However, the significant demand for RF chains tremendously increases the power consumption and the hardware cost, which are commendably reduced by applying DMAs.\n\nIn \\figref{fig:sepSCSIconventional}, we consider four classical kinds of feasible sets of DMA weights, including unconstrained weights, amplitude only, binary amplitude, and Lorentzian phase. As depicted, system SE is achieved to the maximum when DMA weights range at a complex plane, followed by the SE performance of $\\mathrm{SE_{LP}}$, $\\mathrm{SE_{AO}}$, and $\\mathrm{SE_{BA}}$. It is reasonable because the feasible set of unconstrained DMA weights contains the other three conditions. On the conditions of the Lorentzian phase, the optimal value of DMA weights \\iffalse(gains)\\fi is searched in a portion of the complex plane, which makes the optimized SE close to the $\\mathrm{SE_{UC}}$. By comparison, cases amplitude only and binary amplitude optimize DMA weights only on the one-dimensional real axis, which reduces the upper bound of the system SE and achieves almost the same performance. In practice, implementing DMA with binary amplitude has lower hardware costs, which makes it more attractive than the DMA with amplitude only.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.35\\textwidth]{sepSCSIconventional.eps}\n\t\\caption{Comparison of system SE between the DMA assisted and conventional antenna assisted system with partial CSI.}\n\t\\label{fig:sepSCSIconventional}\n\\end{figure}\n\n\\subsection{Comparison With Baseline Approaches}\nTo further demonstrate the effectiveness of the active design that considers EM exposure constraints, we compare the system SE of the proposed algorithm with baseline approaches in \\figref{fig:sepSCSIconventional}, which contain worst-case power backoff and adaptive backoff approach. Both two baseline approaches only consider the power constraints in $\\mathcal{P}_3^a$, i.e., \\eqref{p3c} is not taken into account, and then make the solution in $\\mathcal{P}_3^a$ satisfy the SAR constraints by introducing the backoff factor $\\rho$. The specific process of the two baseline approaches is detailed as follows:\n\\begin{itemize}\n\t\\item \\emph{Worst-Case Power Backoff} \\cite{YLH15}: Denote the worst-case power backoff factor as \n\t\\begin{align}\\label{equ:WCbackoff2}\n\t\\rho_1=\\min\\left\\{1,D\/\\mathrm{SAR^{worst}}\\right\\},\n\t\\end{align}\n\twhere $\\mathrm{SAR^{worst}}$ is the worst SAR given by\n\t\\begin{align}\\label{equ:WCbackoff3}\n\t\\mathrm{SAR^{worst}}=\\underset{k}\\max\\underset{\\tr{\\mathbf{Q}_k}\\leqP_{\\mathrm{max},k}}\\max \\tr{\\mathbf{R}\\mathbf{Q}_k}.\n\t\\end{align}\n\tThis approach only considers the power constraints and then makes the final result satisfy the SAR constraints by reducing the power budgets as \n\t\\begin{align}\\label{equ:WCbackoff1}\n\t\\mathbf{Q}^{\\mathrm{wc}}_{\\mathrm{opt}}=\\argmax{\\tr{\\mathbf{Q}_k}\\leq\\rho_1P_{\\mathrm{max},k}}\\ {\\overline {\\eta}}_{\\mathrm{SE}}(\\mathbf{Q}), \\quad \\forall k.\n\t\\end{align}\n\n\t\\item \\emph{Adaptive Backoff} \\cite{YLH15}: By omitting the SAR constraints, problem $\\mathcal{P}_3^a$ degenerates into\n\t\\begin{align}\\label{equ:Adpbackoff1}\n\t\\mathbf{Q}^{\\mathrm{0}}=\\argmax{\\tr{\\mathbf{Q}_k}\\leqP_{\\mathrm{max},k}}\\ {\\overline {\\eta}}_{\\mathrm{SE}}(\\mathbf{Q}), \\quad \\forall k.\n\t\\end{align}\n\tThen, the adaptive backoff factor $\\rho_2$ is introduced to make the result meet the SAR constraints.\n\t\\begin{align}\\label{equ:Adpbackoff2}\n\t\\rho_{2(k)}=\\min\\left\\{1,D\/\\tr{\\mathbf{R}\\mathbf{Q}^{\\mathrm{0}}_{{k}}}\\right\\},\\quad \\forall k.\n\t\\end{align}\n\tFinally, the corresponding transmit covariance can be expressed as\n\t\\begin{align}\\label{equ:Adpbackoff3}\n\t\\mathbf{Q}^{\\mathrm{adp}}_{k,\\mathrm{opt}}=\\rho_{2(k)}\\mathbf{Q}_k^{\\mathrm{0}},\\ \\forall k.\n\t\\end{align}\n\\end{itemize}\nIt can be observed that when $P_{\\mathrm{max}}$ is lower than a certain threshold, the three methods achieve the same SE performance. Under this condition, SAR constraints can be naturally satisfied even when the power budget is fully used to transmit signals, which means \\eqref{p2ac} or \\eqref{p3c} is actually negligible, and all the backoff factors $\\rho_1$, $\\rho_{2(k)}, \\forall k$ are equal to one. When $P_{\\mathrm{max}}$ is higher than a threshold, SAR constraints start to restrict the increase of system SE in a different way from power constraints. Since the backoff methods only consider the SAR constraint as the additional power constraint and ignore the correlation between SAR and the phase differences of transmit antennas, our proposed method, which considers both the power and SAR constraints, can achieve higher SE.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.35\\textwidth]{sepSCSIbackoff.eps}\n\t\\caption{SE performance comparison between the proposed and baseline approaches with partial CSI.}\n\t\\label{fig:sepSCSIbackoff}\n\\end{figure}\n\n\n\n\\section{Conclusion}\\label{sec:conclusion}\nTo summarize, we investigated the EM exposure constrained uplink strategy design of the hybrid RIS and DMA assisted multiuser MIMO system. We considered a practical scenario where instantaneous CSI is always available for the RIS-to-DMAs channel and discussed the instantaneous and statistical CSI of users-to RIS channel, respectively. To obtain the maximal SE under full or partial CSI, we proposed the corresponding algorithm to jointly optimize the transmit covariance matrices, RIS phase shift matrix, and DMA weight matrix, where power and SAR constraints were included in the optimization. Especially, for the reduction of the computational complexity under partial CSI, we adopted the DEs to asymptotically approximate the ergodic SE and then proposed a SE maximization algorithm that is applicable for partial CSI of users-to-RIS channels. In the numerical simulations, we verified the convergence performance of the overall algorithm and figured out the impact of EM exposure, RIS and DMA on the system SE by comparing with conventional systems. Then, numerical results substantiated the superior SE performance of the EM aware joint optimization design over the backoff approaches.\n\n\n\n\n\n\\appendices\n\n\\section{Proof of \\propref{Prop:bPhi_optimal2}}\\label{app:A}\nAccording to \\cite{MW78}, MM method is an iterative optimization method used to find the approximate solution, where the key is using the local minimizer points of each iteration $\\bm{\\phi}^{(\\ell)}$ to construct a series of upper bound functions $\\widetilde{g}(\\bm{\\phi}|\\bm{\\phi}^{(\\ell)})$ of the original objective $g(\\bm{\\phi})$ satisfying\n\\begin{itemize}\n\t\\item[(i),] $\\widetilde{g}(\\bm{\\phi}|\\bm{\\phi}^{(\\ell)})\\geq g(\\bm{\\phi}),\\quad \\forall \\phi_n, \\phi_n^{(\\ell)}\\in\\mathcal{F}_1$,\n\t\\item[(ii),] $\\widetilde{g}(\\bm{\\phi}^{(\\ell)}|\\bm{\\phi}^{(\\ell)})= g(\\bm{\\phi}^{(\\ell)}),\\quad \\forall\\phi_n^{(\\ell)}\\in\\mathcal{F}_1$,\n\t\\item[(iii),] $\\nabla_{\\bm{\\phi}}\\widetilde{g}(\\bm{\\phi}^{(\\ell)}|\\bm{\\phi}^{(\\ell)})= \\nabla_{\\bm{\\phi}}g(\\bm{\\phi}^{(\\ell)}),\\quad \\forall\\phi_n^{(\\ell)}\\in\\mathcal{F}_1$.\t\n\\end{itemize}\nBased on this condition, the sequence $\\left\\{\\widetilde{g}(\\bm{\\phi}^{(\\ell)}|\\bm{\\phi}^{(\\ell)})\\right\\}_{\\ell=0}^{\\infty}$ is monotonically non-increasing and finally converge to the approximation minimum value of $\\overline{\\mathcal{P}}_{2,\\bm{\\phi}}^{\\mathrm{MSE}}$ with the solution $\\bm{\\phi}_{opt}$ fulfilling the first order Lagrangian conditions. In the following, we prove that the function constructed in \\emph{\\propref{Prop:bPhi_optimal2}} satisfies the above conditions.\n\nNotice $\\bm{\\Delta}$ is a positive semidefinite matrix as $\\mathbf{A}$ and $\\mathbf{P}$ are both positive semidefinite, so is $\\lambda_{\\mathrm{max}}\\mathbf{I}_{N_R}-\\bm{\\Delta}$. Applying that $\\left|\\left|\\left(\\lambda_{\\mathrm{max}}\\mathbf{I}_{N_R}-\\bm{\\Delta}\\right)^{\\frac{1}{2}}\\bm{\\phi}-\\left(\\lambda_{\\mathrm{max}}\\mathbf{I}_{N_R}-\\bm{\\Delta}\\right)^{\\frac{1}{2}}\\bm{\\phi}^{(\\ell)}\\right|\\right|^2\\geq 0$, i.e.,\n\\begin{align}\\label{equ:appendix1a}\n&\\bm{\\phi}^H\\left(\\lambda_{\\mathrm{max}}\\mathbf{I}_{N_R}-\\bm{\\Delta}\\right)\\bm{\\phi}+(\\bm{\\phi}^{(\\ell)})^H\\left(\\lambda_{\\mathrm{max}}\\mathbf{I}_{N_R}-\\bm{\\Delta}\\right)\\bm{\\phi}^{(\\ell)}\\notag\\\\\n&\\qquad\\qquad-2\\Re\\left\\{\\bm{\\phi}^H(\\lambda_{\\mathrm{max}}\\mathbf{I}_{N_R}-\\bm{\\Delta})\\bm{\\phi}^{(\\ell)}\\right\\}\\geq 0,\n\\end{align} \nwe obtain the upper bound function\n\\begin{align}\\label{equ:appendix1b}\n&g(\\bm{\\phi})\\leq\\widetilde{g}(\\bm{\\phi}|\\bm{\\phi}^{(\\ell)})=2\\lambda_{\\mathrm{max}}\\mathbf{I}_{N_R}-(\\bm{\\phi}^{(\\ell)})^H\\bm{\\Delta}\\bm{\\phi}^{(\\ell)}\\notag\\\\\n&\\quad-2\\Re\\left\\{\\bm{\\phi}^H(\\lambda_{\\mathrm{max}}\\mathbf{I}_{N_R}-\\bm{\\Delta})\\bm{\\phi}^{(\\ell)}\\right\\}-2\\Re\\{\\bm{\\phi}^H\\mathbf{b}^*\\}.\n\\end{align} \nIn addition, the identities in conditions (ii) and (iii) also hold referring to \\cite{HZA19}. As the properties of MM method, we aims to address the subproblem iteratively as follows:\n\\begin{subequations}\\label{eq:problem2MMdedu}\n\t\\begin{align}\n\t\\overline{\\mathcal{P}}^{(\\ell)}:\\ \\bm{\\phi}^{(\\ell+1)}=\\argmin{\\bm{\\phi}}\\  &\\widetilde{g}(\\bm{\\phi}|\\bm{\\phi}^{(\\ell)}), \\label{pMMadedu}\\\\\n\t{\\mathrm{s.t.}}\\ &\\phi_n \\in \\mathcal{F}_1,\\quad \\forall n, \\label{pMMbdedu}\n\t\\end{align}\n\\end{subequations}\nwhich is equivalent to $\\overline{\\mathcal{P}}_{2,\\bm{\\phi}}^{(\\zeta)}$. This concludes the proof.\n\n\\section{Proof of \\propref{Prop:Qk_optimal2}}\\label{app:B}\nBy approximating ${\\eta}_{\\mathrm{SE}}(\\mathbf{Q})$ as ${\\overline {\\eta}}_{\\mathrm{SE}}(\\mathbf{Q})$ in \\eqref{equ:Lagrangian1}, we can obtain the Lagrange dual function of $\\mathcal{P}_3^a$ represented by $\\mathcal{L}(\\mathbf{Q})$. Thus the dual problem is expressed by \\eqref{equ:dual_problem1}. As shown in \\cite{WJW11}, the derivative of ${\\overline {\\eta}}_{\\mathrm{SE}}(\\mathbf{Q})$ in \\eqref{equ:DEsGk} over  $\\mathbf{Q}_k$ is given by\n\\begin{align}\\label{equ:partialSE}\n\\frac{\\partial{\\overline {\\eta}}_{\\mathrm{SE}}(\\mathbf{Q})}{\\partial\\mathbf{Q}_k}=(\\sigma^{2}\\mathbf{I}_{N_k}+\\bm{\\Gamma}_{k}^a\\mathbf{Q}_k)^{-1}\\bm{\\Gamma}_{k}^a.\n\\end{align}\nConsider the Karush-Kuhn-Tucker (KKT) conditions:\n\\begin{align}\\label{equ:KKT}\n\\frac{\\partial\\mathcal{L}}{\\partial\\mathbf{Q}_{k,\\mathrm{opt}}}=(\\sigma^{2}\\mathbf{I}_{N_k}+\\bm{\\Gamma}_{k}^a\\mathbf{Q}_{k,\\mathrm{opt}})^{-1}\\bm{\\Gamma}_{k}^a-\\mathbf{K}_{k,\\mathrm{opt}}=\\mathbf{m}{0}.\n\\end{align}\nTherefore, with fixed $\\bm{\\Gamma}_{k}^a$ and given $\\mu_k,\\ \\lambda_{k,i}\\ \\forall k,i$, we can obtain the equivalent inner optimization of \\eqref{equ:dual_problem1} as\n\\begin{align}\\label{equ:innermaxL}\n\\underset{\\mathbf{Q}_k}\\max \\ \\log\\det\\left(\\mathbf{I}_{N_k}+\\frac{1}{\\sigma^{2}}\\bm{\\Gamma}_{k}^a\\mathbf{Q}_k\\right)-\\tr{\\mathbf{K}_{k,\\mathrm{opt}}\\mathbf{Q}_k}.\n\\end{align}\nThen, the optimal $\\mathbf{Q}_k$ can be obtained as \\eqref{Optimal_Q1} referring to the proof of \\cite[Theorem 3.6]{YLH15}, thus omitted here. This concludes the proof.\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n One of the greatest features of topology is the plenty of powerful computational tools. The Serre spectral sequence associated to a fibration is a notable example that allows to extract information about the cohomology of the total space, once known the cohomology of the base and of the fiber. As a particular case, when the fiber is a sphere, the Serre spectral sequence gives back the Gysin long exact sequence associated to a sphere bundle.\n \nThe development of motivic homotopy theory made it possible to efficiently import topological techniques into the algebro-geometric world. This translation is usually not straightforward nor even univoque (e.g. in motivic homotopy theory we have two classifying spaces, the Nisnevich and the \\'etale, unlike the topological picture). Most of the times, in fact, it is not clear how the topologically inspired tool should look like in the motivic world and what assumptions are needed in order to make the translation successful.\n \n In this paper, we would like to propose the construction of a Serre-type spectral sequence for motivic cohomology. This spectral sequence arises from a Postnikov system in a certain triangulated category of motives associated to a morphism of bisimplicial schemes whose fiber is motivically cellular, i.e. a direct sum of Tate motives. As in topology, our Serre spectral sequence reconstructs the motivic cohomology of the total space out of the motivic cohomology of the base and the cellular structure of the fiber. It is a natural generalisation of the Gysin long exact sequence that was first introduced by Smirnov and Vishik in \\cite{smirnov.vishik} for computing the motivic cohomology of the Nisnevich classifying space of othogonal groups, and subsequently studied by the author in \\cite{tanania.b} for the general case of morphisms of simplicial schemes with reduced Tate fiber (the motivic analogues of sphere bundles). For constructing our spectral sequence, we need to work with bisimplicial schemes instead of simplicial ones. Indeed, we want to allow simplicial fibers that are in some sense infinite-dimensional.\n \n The Gysin long exact sequence for motivic cohomology was successfully exploited for computing the motivic cohomology ring of the Nisnevich classifying space $BG$ for some linear algebraic groups $G$, which in turn produced new subtle invariants for $G$-torsors. These new invariants, unlike the usual ones coming from the motivic cohomology of the \\'etale classifying space $B_{\\acute et}G$, take values in a more informative object, namely the motivic cohomology of the \\v{C}ech simplicial scheme of the torsor under study.\n \n More precisely, the Gysin sequence was used for orthogonal groups in \\cite{smirnov.vishik}, for unitary groups in \\cite{tanania.a} and for spin groups in \\cite{tanania.b}. Unfortunately, some algebraic groups are out of its range of application, e.g. the projective general linear group $PGL_n$. The cohomology of its classifying space is notoriously difficult to study even in topology where, neverthless, some important results are known thanks to the use of the Serre spectral sequence. For example, the singular cohomology of $BPU_n$ has been computed in low degrees by Antieau and Williams in \\cite{antieau.williams} and by Gu in \\cite{gu1}. Moreover, the singular cohomology groups and the Chow groups of the \\'etale classifying space of $PGL_p$ for an odd prime $p$ were completely computed by Vistoli in \\cite{vistoli}. The Nisnevich classifying space of $PGL_n$ was first studied by Rolle in \\cite{rolle}. In this paper, we show how to apply our spectral sequence to obtain some information about the motivic cohomology of $BPGL_n$. Indeed, we use the Serre spectral sequence developed here for the obvious map $BGL_n \\rightarrow BPGL_n$ with fiber $B\\Gm$ in order to compute the motivic cohomology of $BPGL_n$ in low motivic weights.\n \nAs noted before, investigating classifying spaces from a cohomological point of view is essential for the program of classifying torsors of linear algebraic groups. For example, the importance of studying Nisnevich classifying spaces for the classification of quadratic forms was highlighted by Smirnov and Vishik in \\cite{smirnov.vishik}. In the case of projective general linear groups, torsors are central simple algebras and a better understanding of the Nisnevich classifying space of $PGL_n$ provides information about them. \n\nMotivic descriptions of geometric objects related to central simple algebras have been widely investigated in the last decades. For example, Karpenko shows in \\cite{karpenko} that the Chow motive of the Severi-Brauer variety associated to a central simple algebra is the direct sum of twisted copies of the motive of the Severi-Brauer variety associated to the underlying division algebra, while the latter is indecomposable. In \\cite{kahn.levine}, Kahn and Levine otbain a Postnikov tower in the Voevodsky's triangulated category of motives for the Severi-Brauer variety associated to a division algebra of prime degree. In \\cite{shinder}, Shinder studies the slice filtration for the motive of the group of units of a division algebra of prime degree, which involves the motive of the associated \\v{C}ech simplicial scheme. In this paper, we show that the Postnikov system giving rise to our Serre spectral sequence for $BPGL_n$ also provides a description of the motive of a Severi-Brauer variety associated to any central simple algebra in terms of the respective \\v{C}ech simplicial scheme. The latter description also induces a spectral sequence, whose first differential is fully understood, strongly converging to the motivic cohomology of the Severi-Brauer variety starting from the motivic cohomology of its \\v{C}ech simplicial scheme.\n\nAs we have already pointed out, a complete description of the singular cohomology of $BPU_n$ seems to be still out of reach. Neverthless, in \\cite{gu2} and \\cite{gu3} Gu has constructed non-trivial torsion classes in the cohomology ring of $BPU_n$ and in the Chow ring of $B_{\\acute et}PGL_n$ over the complex numbers, by using Steenrod operations and the cycle class map going from Chow groups to singular cohomology. These important results shed new light on the structure of these rings. In this paper, by following the lead of \\cite{gu2} and \\cite{gu3}, we show that there are similar non-trivial torsion classes in the motivic cohomology of the Nisnevich classifying space of $PGL_n$. By exploiting the homomorphism in motivic cohomology induced by the canonical map $BPGL_n \\rightarrow B_{\\acute et}PGL_n$ (that replaces the cycle class map over general fields), we then generalise Gu's results on the torsion classes in the Chow ring of $B_{\\acute et}PGL_n$ to fields of characteristic not dividing $n$ containing a primitive $n$th root of unity.\\\\\n\n\\textbf{Outline.} In Section 2, we report the main notations we need in this paper. In Section 3, we recall a few things about Postnikov systems in triangulated categories and induced spectral sequences. Section 4 is devoted to the introduction of the triangulated category of motives over a bisimplicial scheme that is modeled on the one constructed for simplicial schemes by Voevodsky in \\cite{voevodsky.simplicial}. Section 5 contains the main result of this paper, i.e. the construction of the Serre spectral sequence for motivic cohomology (see Theorem \\ref{Serre2}). In Section 6, we apply the latter to compute the motivic cohomology of the Nisnevich classifying space of $PGL_n$ in low motivic weights (see Theorem \\ref{bpg}). Section 7 provides a Postnikov system for the motive of a Severi-Brauer variety (see Proposition \\ref{pssb}) induced by the one for the map $BGL_n \\rightarrow BPGL_n$ studied in Section 6. Finally, in Section 8, we find torsion classes in the motivic cohomology of $BPGL_n$ generalising some results from \\cite{gu2} and \\cite{gu3} (see Corollary \\ref{gentor}).\\\\\n\n\\textbf{Acknowledgements.} I would like to thank Alexander Vishik and Olivier Haution for helpful conversations on the topic of this paper.\n \n \\section{Notation}\n\nHere we fix some notations we use throughout this paper.\\\\\n\n\\begin{tabular}{c|c}\n\t$k$ & perfect field\\\\\n\t$R$ & commutative ring with identity\\\\\n\t$Y_{\\bullet,\\bullet}$ & smooth bisimplicial scheme over $k$\\\\\n\t$d(Y_{\\bullet,\\bullet})$ & diagonal simplicial scheme of $Y_{\\bullet,\\bullet}$\\\\\n\t$\\DM^-_{eff}(k,R)$ & triangulated category of motives over $k$ with $R$-coefficients \\\\\n\t$\\DM^-_{eff}(Y_{\\bullet,\\bullet},R)$ & triangulated category of motives over $Y_{\\bullet,\\bullet}$ with $R$-coefficients\\\\\n\t$\\DM_{coh}^-(Y_{\\bullet,\\bullet},R)$ & localizing subcategory of coherent motives over $Y_{\\bullet,\\bullet}$ with $R$-coefficients\\\\\n\t$H^{**}(-,R)$ & motivic cohomology with $R$-coefficients\\\\\n\t$H^{**}(-)$ & motivic cohomology with $\\Z$-coefficients\\\\\n\t$CH^*(-)$ & Chow groups, i.e. $CH^q(-) \\cong H^{2q,q}(-)$\\\\\n\t${\\mathcal H}_s(k)$ & simplicial homotopy category over $k$\\\\\n\t${\\mathcal H}(k)$ & $A^1$-homotopy category over $k$\\\\\n\t$BG_{\\bullet}$ & Nisnevich classifying space of the simplicial algebraic group $G_{\\bullet}$\\\\\n\t$B_{\\acute et}G$ & \\'etale classifying space of the algebraic group $G$\\\\\n\t$PGL_n$ & projective general linear group\\\\\n    $A$ & central simple algebra over $k$\\\\\n    $\\seb(A)$ & Severi-Brauer variety associated to $A$\\\\\n    $\\X_A$ & motive of the \\v{C}ech simplicial scheme of $\\seb(A)$\\\\\n\\end{tabular}\\\\\n\n\n\\section{Some general facts about spectral sequences}\\label{spec}\n\nLet us start by recalling some well known facts about Postnikov systems in triangulated categories, spectral sequences associated to them and convergence issues (see \\cite{boardman}, \\cite{gelfand.manin} and \\cite{mccleary}).\n\nThroughout this section, we denote by $\\mathcal{C}$ a triangulated category, by $\\A$ an abelian category and by $H:\\mathcal{C} \\rightarrow \\A$ a cohomological functor.\n\n\\begin{dfn}\\label{ec}\n\tAn exact couple in $\\A$ is a triangle\n\t$$\n\t\\xymatrix{\n\t\tD  \\ar@{->}[rr]^{i} & & D \\ar@{->}[dl]^{j}\\\\\n\t\t& E \\ar@{->}[ul]^{k}&\n\t}\n\t$$\n\twhich is exact at each vertex.\n\\end{dfn}\n\nThe morphism $d$ defined as the composition $jk$ is a differential, i.e. $d^2=0$. Set $E'=\\ker(d)\/\\Ima(d)$ and $D'=\\Ima(i)$. By some standard arguments, one can define morphisms $j':D' \\rightarrow E'$ and $k':E' \\rightarrow D'$ respectively by $j'(i(x))=j(x)$ and $k'([y])=k(y)$ for any $x \\in D$ and $y \\in \\ker(d)$. The triangle obtained in this way\n$$\n\\xymatrix{\n\tD'  \\ar@{->}[rr]^{i} & & D' \\ar@{->}[dl]^{j'}\\\\\n\t& E' \\ar@{->}[ul]^{k'}&\n}\n$$\nis again an exact couple, called the derived couple. Reiterating this construction, one gets a sequence of objects $E_r$ in $\\A$, endowed with differentials $d_r$, each of which is the homology of the previous one. More precisely, we can give the following definition.\n\n\\begin{dfn}\n\tThe sequence $\\{E_r,d_r\\}_{r \\geq 1}$ constructed inductively by\n\t$$E_r=\\ker(d_{r-1})\/\\Ima(d_{r-1})$$\n\tis called the spectral sequence associated to the exact couple in Definition \\ref{ec}.\n\\end{dfn}\n\nIn practical situations, one often encounters bigraded exact couples, which naturally give rise to bigraded spectral sequences $\\{E_r^{s,t},d_r^{s,t}\\}_{r \\geq 1}$. We now provide the definition of the limit page of a bigraded spectral sequence.\n\n\\begin{dfn}\n\tLet $\\{E_r^{s,t},d_r^{s,t}\\}_{r \\geq 1}$ be a bigraded spectral sequence and suppose there is an integer $r(s,t)$ such that $E_r^{s,t} \\cong E_{r(s,t)}^{s,t}$ for any $r \\geq r(s,t)$, then we say that the spectral sequence abuts to $E_{\\infty}^{s,t}=E_{r(s,t)}^{s,t}$.\n\\end{dfn}\n\nAt this point, let us recall some notions about filtrations and covergence of spectral sequences from \\cite{boardman}.\n\nAn increasing filtration of an object $G$ in $\\A$ is a diagram of the following shape \n$$\\dots \\hookrightarrow F^1 \\hookrightarrow F^2 \\hookrightarrow \\dots \\hookrightarrow F^m \\hookrightarrow F^{m+1} \\hookrightarrow \\dots \\hookrightarrow G.$$\n\n\\begin{dfn}\n\tThe increasing filtration $\\{F^m\\}_{m \\in \\Z}$ of  $G$ is said to be:\\\\\n\t1) exhaustive if $G=\\varinjlim_m F^m$;\\\\\n\t2) Hausdorff if $\\varprojlim_m F^m=0$;\\\\\n\t3) complete if $\\varprojlim^1_m F^m=0$.\n\\end{dfn}\n\nIn practice, one is often interested in filtrations of graded objects. So, for any $F^s$ in the filtration we denote by $F^{s,t}$ its graded component in degree $t$.\n\n\\begin{dfn}\n\tA spectral sequence associated to an exact couple is called:\\\\\n\t1) weakly convergent to $G$ if there exists an increasing filtration of $G$ which is exhaustive and such that $E^{s,t}_{\\infty} \\cong F^{s,t}\/F^{s-1,t}$ for any $s$;\\\\\n\t2) convergent to $G$ if it is weakly convergent and the filtration of $G$ is Hausdorff;\\\\\n\t3) strongly convergent to $G$ if it is weakly convergent and the filtration of $G$ is complete Hausdorff.\n\\end{dfn}\n\nWe now recall the definition of Postnikov system in a triangulated category (see \\cite{gelfand.manin}). \n\n\\begin{dfn}\\label{ps}\n\tA Postnikov system for an object $X$ in $\\mathcal{C}$ is a diagram\n\t$$\n\t\\xymatrix{\n\t\t\\dots \\ar@{->}[r] & X_{i+1} \\ar@{->}[r] \\ar@{->}[d]\t &X_i \\ar@{->}[r] \\ar@{->}[d] &  \\dots \\ar@{->}[r]   & X_2 \\ar@{->}[r] \\ar@{->}[d]\t & X=X_1  \\ar@{->}[d]  \\\\\n\t\t& Y_{i+1} \\ar@{->}[ul]^{[1]} &\tY_i \\ar@{->}[ul]^{[1]} & & Y_2 \\ar@{->}[ul]^{[1]}  &\tY_1 \\ar@{->}[ul]^{[1]} \n\t}\n\t$$\n\twhere all the triangles are distinguished triangles in $\\mathcal{C}$.\n\\end{dfn}\n\nAssociated to a Postnikov system one can always construct an exact couple by applying a cohomological functor $H$. More precisely, we have the following bigraded exact couple\n$$\n\\xymatrix{\n\tD  \\ar@{->}[rr]^{i} & & D \\ar@{->}[dl]^{j}\\\\\n\t& E \\ar@{->}[ul]^{k}&\n}\n$$\nwhere $D^{s,t}=H(X_s[-t])$, $E^{s,t}=H(Y_s[-t])$ and the morphisms $i:D^{s,t} \\rightarrow D^{s+1,t}$, $j:D^{s,t} \\rightarrow E^{s-1,t+1}$ and $k:E^{s,t} \\rightarrow D^{s,t}$ are induced by the morphisms in the Postnikov system.\n\nAs usual, we obtain an increasing filtration of the object $H(X)$ in $\\A$ given by \n$$F^1 \\hookrightarrow F^2 \\hookrightarrow \\dots \\hookrightarrow F^m \\hookrightarrow F^{m+1} \\hookrightarrow \\dots \\hookrightarrow H(X)$$\nwhere $F^m=\\ker(H(X) \\rightarrow H(X_m))$ and the morphism $H(X) \\rightarrow H(X_m)$ is the one induced by the Postnikov system. Moreover, observe that the filtration $\\{F^m\\}_{m \\geq 1}$ just introduced is complete Hausdorff, since it is bounded from below, but not necessarily exhaustive.\tAnyway, we have the following result that guarantees the strong convergence of the spectral sequence just constructed, provided that a certain condition holds.\n\n\\begin{thm}\\label{SC}\n\tIf $\\varinjlim_m H(X_m) \\cong 0$, then the spectral sequence associated to the Postnikov system in Definition \\ref{ps} is strongly convergent to $H(X)$.\n\\end{thm}\n\\begin{proof}\n\tSee \\cite[Theorem 6.1]{boardman}.\n\\end{proof}\n\n\\section{Motives over a bisimplicial scheme}\n\nFor technical reasons, in this paper we need to work over bisimplicial schemes. To this end, we need a triangulated category of motives over a bisimplicial scheme. The triangulated category of motives over a simplicial scheme was introduced and studied in \\cite{voevodsky.simplicial}. We point out that the constructions and results we need from \\cite{voevodsky.simplicial} extend to the bisimplicial case in a straightforward way. In this section we briefly summarise them. \n\nLet $Y_{\\bullet,\\bullet}$ be a smooth bisimplicial scheme over $k$. Following \\cite[Section 2]{voevodsky.simplicial}, we define $Sm\/Y_{\\bullet,\\bullet}$ in the following way.\n\n\\begin{dfn}\nDenote by $Sm\/Y_{\\bullet,\\bullet}$ the category whose objects are triples $(U,i,h)$, where $i$ and $h$ are non-negative integers and $U$ is a smooth scheme over $Y_{i,h}$, and whose morphisms from $(U,i,h)$ to $(V,j,k)$ are triples $(f,\\phi,\\psi)$, where $\\phi:[j] \\rightarrow [i]$ and $\\psi:[k] \\rightarrow [h]$ are simplicial maps and $f:U \\rightarrow V$ is a morphism of schemes such that the square\n$$\n\\xymatrix{\n\tU \\ar@{->}[r]^{f} \\ar@{->}[d] & V \\ar@{->}[d]\\\\\n\tY_{i,h} \\ar@{->}[r]_{Y_{\\phi,\\psi}} & Y_{j,k}\n}\n$$\ncommutes.\n\\end{dfn}\n\nWe can also define presheaves on $Y_{\\bullet,\\bullet}$ following \\cite[Definition 2.1]{voevodsky.simplicial}.\n\n\\begin{dfn}\n\tA presheaf of sets (respectively with transfers) on $Y_{\\bullet,\\bullet}$ consists of a collection $\\{F_{i,h}\\}_{i,h \\geq 0}$ of presheaves of sets (respectively with transfers) on $Sm\/Y_{i,h}$ together\n\twith a morphism of presheaves of sets (respectively with transfers) $F_{\\phi,\\psi}: Y_{\\phi,\\psi}^*(F_{j,k}) \\rightarrow  F_{i,h}$ for any simplicial maps $\\phi:[j] \\rightarrow [i]$ and $\\psi:[k] \\rightarrow [h]$, such that $F_{id,id} =id$ and $F_{\\phi\\alpha,\\psi\\beta} :Y_{\\phi\\alpha,\\psi\\beta}^*(F_{m,n}) \\rightarrow F_{i,h}$ is equal to the composition of $Y_{\\phi,\\psi}^*F_{\\alpha,\\beta} : Y_{\\phi,\\psi}^*Y_{\\alpha,\\beta}^*(F_{m,n})  \\rightarrow  Y_{\\phi,\\psi}^*(F_{j,k})$ and $F_{\\phi,\\psi} : Y_{\\phi,\\psi}^*(F_{j,k})  \\rightarrow  F_{i,h}$, where $\\alpha : [m]  \\rightarrow  [j]$ and $\\beta : [n]  \\rightarrow  [k]$ are simplicial maps.\n\\end{dfn}\n\nDenote by $PShv(Y_{\\bullet,\\bullet})$ the category of presheaves of sets on $Y_{\\bullet,\\bullet}$ and by $PST(Y_{\\bullet,\\bullet},R)$ the abelian category of presheaves with transfers on $Y_{\\bullet,\\bullet}$.\nNote that $PShv(Y_{\\bullet,\\bullet})$ is nothing but the category of contravariant functors from $Sm\/Y_{\\bullet,\\bullet}$ to $Sets$.\n\nIf $F = \\{F_{i,h}\\}_{i,h \\geq 0}$ is a presheaf of sets on $Y_{\\bullet,\\bullet}$, then $R_{tr}F = \\{R_{tr}F_{i,h}\\}_{i,h \\geq 0}$ is a presheaf with transfers on $Y_{\\bullet,\\bullet}$. In particular, denote by $R_{tr}(U,i,h)$ the presheaf with transfers associated to the representable presheaf of sets corresponding to $(U,i,h)$.\n\nLet $SmCor(Y_{\\bullet,\\bullet},R)$ be the full subcategory of $PST(Y_{\\bullet,\\bullet},R)$ whose objects are direct sums of objects of the form $R_{tr}(U,i,h)$.\n\n\\begin{lem}\n\tThe category $PST(Y_{\\bullet,\\bullet},R)$ is naturally equivalent to the category of $R$-linear contravariant functors from $SmCor(Y_{\\bullet,\\bullet},R)$ to the category of $R$-modules that preserve coproducts.\n\\end{lem}\n\\begin{proof}\n\tSee \\cite[Lemma 2.3]{voevodsky.simplicial}.\n\t\\end{proof}\n\nThe previous result allows, as usual, to construct left resolutions $Lres(F)$ consisting of representable presheaves with transfers for any $F$ in $PST(Y_{\\bullet,\\bullet},R)$.\n\nFor any non-negative integers $i$ and $h$ denote by $r_{i,h} : SmCor(Y_{i,h},R)  \\rightarrow SmCor(Y_{\\bullet,\\bullet},R)$ the functor that sends $U$ to $R_{tr}(U,i,h)$. These functors induce in the standard way pairs of adjoint functors\n\\begin{align*}\n\tPST&(Y_{\\bullet,\\bullet},R)\\\\\n\tr_{i,h,\\#} \\uparrow & \\downarrow r_{i,h}^*\\\\\n\tPST&(Y_{i,h},R).\n\\end{align*}\n\nThere are similar functors $r_{i,\\bullet} : SmCor(Y_{i,\\bullet},R)  \\rightarrow SmCor(Y_{\\bullet,\\bullet},R)$ sending $R_{tr}(U,h)$ to $R_{tr}(U,i,h)$, which induce pairs of adjoint functors\n\\begin{align*}\n\tPST&(Y_{\\bullet,\\bullet},R)\\\\\n\tr_{i,\\bullet,\\#} \\uparrow & \\downarrow r_{i,\\bullet}^*\\\\\n\tPST&(Y_{i,\\bullet},R),\n\\end{align*}\n\nand $r_{\\bullet,h} : SmCor(Y_{\\bullet,h},R)  \\rightarrow SmCor(Y_{\\bullet,\\bullet},R)$ sending $R_{tr}(U,i)$ to $R_{tr}(U,i,h)$, which induce pairs of adjoint functors\n\\begin{align*}\n\tPST&(Y_{\\bullet,\\bullet},R)\\\\\n\tr_{\\bullet,h,\\#} \\uparrow & \\downarrow r_{\\bullet,h}^*\\\\\n\tPST&(Y_{\\bullet,h},R).\n\\end{align*}\n\nFinally, we can also consider the diagonal functor $d : SmCor(d(Y_{\\bullet,\\bullet}),R)  \\rightarrow SmCor(Y_{\\bullet,\\bullet},R)$ that sends $R_{tr}(U,i)$ to $R_{tr}(U,i,i)$. As usual, this functor induces a pair of adjoint functors\n\\begin{align*}\n\tPST&(Y_{\\bullet,\\bullet},R)\\\\\n\td_{\\#} \\uparrow & \\downarrow d^*\\\\\n\tPST&(d(Y_{\\bullet,\\bullet}),R).\n\\end{align*}\n\nAs in \\cite[Section 3]{voevodsky.simplicial}, we can define the tensor product of presheaves with transfers $F$ and $G$ on $Y_{\\bullet,\\bullet}$ in the following way\n$$(F \\otimes G)_{i,h} = (F_{i,h} \\otimes G_{i,h}) = h_0(Lres(F_{i,h}) \\otimes Lres(G_{i,h} )).$$\nLet $D(Y_{\\bullet,\\bullet},R)$ be the derived category of complexes on $PST(Y_{\\bullet,\\bullet},R)$ bounded from above. Then, the tensor product just defined induces a tensor triangulated structure on $D(Y_{\\bullet,\\bullet},R)$ given by\n$$K \\stackrel{L}{\\otimes} L = Lres(K) \\otimes Lres(L)$$\nfor all complexes of presheaves with transfers $K$ and $L$.\n\nThe unit of this tensor structure is the constant presheaf with transfers denoted also by $R$ whose components are the constant presheaves with transfers on each $Y_{i,h}$.\n\n\\begin{lem}\\label{tens}\n\tConsider the bisimplicial object $LR_{\\bullet,\\bullet}$ in $SmCor(Y_{\\bullet,\\bullet},R)$ with terms \n\t$$LR_{i,h} =R_{tr}(Y_{i,h},i,h)$$\n\tand the obvious structure morphisms. Let $LR_*$ be the total complex of the corresponding double complex. Then there is a natural quasi-isomorphism\n\t$$LR_* \\rightarrow R.$$\n\\end{lem}\n\\begin{proof}\n\tSee \\cite[Lemma 3.9]{voevodsky.simplicial}.\n\t\\end{proof}\n\nLet $W_{i,h}^{el}(Y_{\\bullet,\\bullet},R)$ be the class of complexes on $PST(Y_{\\bullet,\\bullet},R)$ obtained as $r_{i,h,\\#}(W^{el}(Y_{i,h},R))$, where $W^{el}(Y_{i,h},R)$ is defined in \\cite[Section 4]{voevodsky.simplicial}.\n\nDenote by $W(Y_{\\bullet,\\bullet},R)$ the smallest localizing subcategory of $D(Y_{\\bullet,\\bullet},R)$ containing all $W_{i,h}^{el}(Y_{\\bullet,\\bullet},R)$. A morphism in $D(Y_{\\bullet,\\bullet},R)$ is called an $A^1$-equivalence if its cone lives in $W(Y_{\\bullet,\\bullet},R)$.\n\n\\begin{dfn}\nThe triangulated category $\\DM^-_{eff}(Y_{\\bullet,\\bullet},R)$ of motives over $Y_{\\bullet,\\bullet}$ is the localization of $D(Y_{\\bullet,\\bullet},R)$ with respect to $A^1$-equivalences.\n\\end{dfn}\n\nWhat follows consists of a bunch of properties of the restriction functors whose simplicial analogues can be found in \\cite[Sections 3 and 4]{voevodsky.simplicial}.\n\nThe family $\\{r_{i,h}^*\\}_{i,h \\geq 0}$ induces a family of restriction functors from $D(Y_{\\bullet,\\bullet},R)$ to $D(Y_{i,h},R)$, with respective left adjoints $Lr_{i,h,{\\#}}$, that respect $A^1$-equivalences. Hence, we get a family of restriction functors $\\{r_{i,h}^*\\}_{i,h \\geq 0}$ from $\\DM_{eff}^-(Y_{\\bullet,\\bullet},R)$ to $\\DM_{eff}^-(Y_{i,h},R)$ that is moreover conservative. The same is true also for the families of functors $\\{r_{i,\\bullet}^*\\}_{i \\geq 0}$ and $\\{r_{\\bullet,h}^*\\}_{h \\geq 0}$.\n\nThe diagonal functor $d^*$ also induces a functor from $D(Y_{\\bullet,\\bullet},R)$ to $D(d(Y_{\\bullet,\\bullet}),R)$, with left adjoint $Ld_{\\#}$, respecting $A^1$-equivalences. Therefore, we get a diagonal restriction functor $d^*$ from $\\DM_{eff}^-(Y_{\\bullet,\\bullet})$ to $\\DM_{eff}^-(d(Y_{\\bullet,\\bullet}))$.\n\nThe tensor product on $D(Y_{\\bullet,\\bullet},R)$ respects $A^1$-equivalences, making $\\DM^-_{eff}(Y_{\\bullet,\\bullet},R)$ into a tensor triangulated category. All the restriction functors introduced above respect this tensor structure.\n\n There are also standard functoriality properties. If $p:Y_{\\bullet,\\bullet} \\rightarrow Y'_{\\bullet,\\bullet}$ is a morphism of smooth bisimplicial schemes, then we get a pair of adjoint functors\n\\begin{align*}\n\t\\DM_{eff}^-&(Y_{\\bullet,\\bullet},R)\\\\\n\tLp^* \\uparrow & \\downarrow Rp_*\\\\\n\t\\DM_{eff}^-&(Y'_{\\bullet,\\bullet},R).\n\\end{align*}\nIf $p$ is smooth, then $Lp^*=p^*$ and there is also the following adjunction\n\\begin{align*}\n\t\\DM_{eff}^-&(Y_{\\bullet,\\bullet},R)\\\\\n\tLp_{\\#} \\downarrow & \\uparrow p^*\\\\\n\t\\DM_{eff}^-&(Y'_{\\bullet,\\bullet},R).\n\\end{align*}\nIn particular, we have a pair of adjoint functors\n\\begin{align*}\n\t\\DM_{eff}^-&(Y_{\\bullet,\\bullet},R)\\\\\n\tLc_{\\#} \\downarrow & \\uparrow c^*\\\\\n\t\\DM_{eff}^-&(k,R)\n\\end{align*}\nwhere $c:Y_{\\bullet,\\bullet} \\rightarrow Spec(k)$ is the projection to the base. \n\n\\begin{dfn}\n\tA Tate object $T(q)[p]$ in $\\DM^-_{eff}(Y_{\\bullet,\\bullet},R)$ is defined as $c^*(T(q)[p])$. In general, for any motive $M$ in $\\DM^-_{eff}(k,R)$, we also denote by $M$ its image $c^*(M)$ in $\\DM^-_{eff}(Y_{\\bullet,\\bullet},R)$.\n\\end{dfn}\n\nA smooth bisimplicial scheme $Y_{\\bullet,\\bullet}$ induces a bisimplicial object $R_{tr}(Y_{\\bullet,\\bullet})$ in $SmCor(k,R)$. Then, one can define the motive $M(Y_{\\bullet,\\bullet})$ of $Y_{\\bullet,\\bullet}$ in $\\DM^{-}_{eff}(k,R)$ as the total complex of the double complex $R_{tr}(Y_{*,*})$ associated to $R_{tr}(Y_{\\bullet,\\bullet})$. It is an immediate consequence of Lemma \\ref{tens} that $$M(Y_{\\bullet,\\bullet}) \\cong Lc_{\\#}T.$$\nThe latter definition naturally extends to the bisimplicial case the definition of the motive of a simplicial scheme given in \\cite[Section 5]{voevodsky.simplicial}. Note that, by the Eilenberg-Zilber theorem, $M(Y_{\\bullet,\\bullet})$ and $M(d(Y_{\\bullet,\\bullet}))$ are isomorphic in $\\DM^{-}_{eff}(k,R)$. In particular, they have the same motivic cohomology.\n\nThe most important result that we need in the following sections is the following.\n\n\\begin{prop}\n\tLet $Y_{\\bullet,\\bullet}$ be a smooth bisimplicial scheme. Then, there is an isomorphism\n\t$$\\Hom_{\\DM^{-}_{eff}(Y_{\\bullet,\\bullet},R)}(T(q')[p'],T(q)[p]) \\cong \\Hom_{\\DM^{-}_{eff}(k,R)}(M(Y_{\\bullet,\\bullet})(q')[p'],T(q)[p])$$\n\tfor all integers $q$, $q'$, $p$ and $p'$.\n\\end{prop}\n\\begin{proof}\n\tSee \\cite[Proposition 5.3]{voevodsky.simplicial}.\n\t\\end{proof}\n\n\\section{A Serre spectral sequence for motivic cohomology} \n\nThe main purpose of this section is to construct Postnikov systems in a suitable triangulated category of motives and to study the associated spectral sequences. More precisely, we set our triangulated category $\\mathcal{C}$ to be $\\DM^-_{eff}(Y_{\\bullet,\\bullet},R)$, our abelian category $\\A$ to be the category of $H^{**}(Y_{\\bullet,\\bullet},R)$-modules and our cohomological functor $H$ to be motivic cohomology $H^{**}(-,R)$.\n\nFor all $i \\geq 0$ denote simply by\n$$r_i^*:\\DM_{eff}^-(Y_{\\bullet,\\bullet},R) \\rightarrow \\DM_{eff}^-(Y_{i,\\bullet},R)$$\nthe restriction functors $r_{i,\\bullet}^*$ introduced in the last section, and by $Lr_{i,\\#}$ the respective left adjoint functors. The image of a motive $N$ in $\\DM_{eff}^-(Y_{\\bullet,\\bullet},R)$ under $r_i^*$ is simply denoted by $N_i$.\n\nNow let us recall some facts about coherence from \\cite{smirnov.vishik} and adapt them to the bisimplicial case we are interested in. \n\n\\begin{dfn} \n\tA smooth coherent morphism of smooth bisimplicial schemes is a smooth morphism $\\pi:X_{\\bullet,\\bullet} \\rightarrow Y_{\\bullet,\\bullet}$ such that there is a cartesian square of simplicial schemes\n\t$$\n\t\\xymatrix{\n\t\tX_{j,\\bullet} \\ar@{->}[r]^{\\pi_j} \\ar@{->}[d]_{X_ {\\theta}} & Y_{j,\\bullet}  \\ar@{->}[d]^{Y_ {\\theta} }\\\\\n\t\tX_{i,\\bullet}  \\ar@{->}[r]_{\\pi_{i}} & Y_{i,\\bullet} \n\t}\n\t$$\n\tfor any simplicial map $\\theta:[i] \\rightarrow [j]$.\n\\end{dfn} \n\n\\begin{dfn}\n\tA motive $N$ in $\\DM_{eff}^-(Y_{\\bullet,\\bullet},R)$ is said to be coherent if all simplicial morphisms $\\theta:[i] \\rightarrow [j]$ induce structural isomorphisms $N_ \\theta :LY_ {\\theta} ^*(N_i) \\rightarrow N_j$ in $\\DM_{eff}^-(Y_{j,\\bullet},R)$. The full subcategory of $\\DM_{eff}^-(Y_{\\bullet,\\bullet},R)$ consisting of coherent motives is denoted by $\\DM_{coh}^-(Y_{\\bullet,\\bullet},R)$.\n\\end{dfn}\n\n\\begin{rem}\\label{cohloc}\n\t\\normalfont\n Note that $\\DM_{coh}^-(Y_{\\bullet,\\bullet},R)$ is a localizing subcategory of $\\DM_{eff}^-(Y_{\\bullet,\\bullet},R)$. Since $L\\pi_{\\#}$ maps coherent motives to coherent ones for any smooth coherent morphism $\\pi$, we have that $M(X_{\\bullet,\\bullet} \\xrightarrow{\\pi} Y_{\\bullet,\\bullet})$ is an object in $\\DM_{coh}^-(Y_{\\bullet,\\bullet},R)$, where $M(X_{\\bullet,\\bullet} \\xrightarrow{\\pi} Y_{\\bullet,\\bullet})$ is the image $L\\pi_{\\#}(T)$ of the unit Tate motive in $\\DM_{eff}^-(X_{\\bullet,\\bullet},R)$.\n\\end{rem}\n\n\\begin{prop}\\label{filtr}\n\tFor any motive $N$ in $\\DM_{coh}^-(Y_{\\bullet,\\bullet},R)$ there exists a functorial increasing filtration \n\t$$(N)_{\\leq 0} \\rightarrow (N)_{\\leq 1} \\rightarrow \\dots \\rightarrow (N)_{\\leq n-1} \\rightarrow (N)_{\\leq n} \\rightarrow \\dots \\rightarrow N$$\n\twith graded pieces $(N)_n=Cone((N)_{\\leq n-1} \\rightarrow (N)_{\\leq n}) \\cong Lr_{n,\\#}r^*_n(N)[n]$ which converges in the sense that\n\t$$\\bigoplus_n (N)_{\\leq n} \\xrightarrow{id-sh} \\bigoplus_n (N)_{\\leq n} \\rightarrow N$$\n\textends to a distinguished triangle, where $sh:(N)_{\\leq n-1} \\rightarrow (N)_{\\leq n}$ is the map from the filtration.\n\\end{prop}\n\\begin{proof}\n\tThe proofs of \\cite[Propositions 3.1.6 and 3.1.8]{smirnov.vishik} extend verbatim to the bisimplicial case. \n\\end{proof}\n\nThe next proposition is a generalisation of \\cite[Proposition 3.1.5]{smirnov.vishik}. Indeed, it allows to construct Postnikov systems for coherent motives with simplicial components which are direct sums of Tate motives satisfying some specific conditions. The proof follows the guidelines of \\cite[Proposition 3.1.5]{smirnov.vishik} and essentially reproduces the same arguments in our more general context. Before proceeding, we need to define a strict order relation on the bidegrees $(q)[p]$.\n\n\\begin{dfn}\n\tWe set $(q)[p] \\prec (q')[p']$ if and only if one of the following two conditions is satisfied:\\\\\n\t1) $q<q'$;\\\\\n\t2) $q=q'$ and $p<p'$.\n\\end{dfn}\n\nFor any $j \\geq 0$, let $T^j$ be the possibly infinite direct sum $\\bigoplus_{I_j} T(q_j)[p_j]$ in $\\DM_{eff}^-(k,R)$ such that $(q_j)[p_j] \\prec (q_{j+1})[p_{j+1}]$ and let $N$ in $\\DM_{coh}^-(Y_{\\bullet,\\bullet},R)$ be such a motive that its simplicial components $N_i$ in $\\DM_{eff}^-(Y_{i,\\bullet},R)$ are isomorphic to the direct sum $\\bigoplus_{j \\geq 0} T^j$.\n\nNote that in $\\DM_{eff}^-(Y_{1,\\bullet},R)$ the automorphism group $Aut(\\bigoplus_{j \\geq 0} T^j)$ consists of invertible upper triangular matrices, since\n$$\\Hom_{\\DM_{eff}^-(Y_{1,\\bullet},R)}(T,T(q)[p]) \\cong H^{p,q}(Y_{1,\\bullet},R) \\cong 0$$ \nfor any $(q)[p] \\prec (0)[0]$.\n\nRecall that, since $N$ is coherent, for any simplicial map $\\theta:[i] \\rightarrow [j]$, the structural map $N_{\\theta}:LY^*_{\\theta}(N_i) \\rightarrow N_j$ is an isomorphism. In particular, we have two automorphisms\n$$N_{\\partial_0}:LY^*_{\\partial_0}(N_0) \\cong \\bigoplus_{j \\geq 0} T^j \\rightarrow N_1 \\cong \\bigoplus_{j \\geq 0} T^j$$ \nand\n$$N_{\\partial_1}:LY^*_{\\partial_1}(N_0) \\cong \\bigoplus_{j \\geq 0} T^j \\rightarrow N_1 \\cong \\bigoplus_{j \\geq 0} T^j.$$ \n\n\\begin{dfn}\nDenote by $\\omega^N$ the composition $N_{\\partial_1} \\circ N_{\\partial_0}^{-1}$ belonging to $Aut(N_1)$. For any $j \\geq 0$, consider the homomorphism $Aut(\\bigoplus_{j \\geq 0} T^j) \\rightarrow Aut(T^j)$ that sends each invertible upper triangular matrix to its $j$th diagonal entry. We denote the images of $\\omega^N$ under these homomorphisms by $\\omega^N_j$.\n\\end{dfn}\n\n\\begin{rem}\n\t\\normalfont\n\tNote that, if $Y_{0,\\bullet} \\cong Spec(k)$, then we simply have that $Y_{\\partial_0}=Y_{\\partial_1}:Y_{1,\\bullet} \\rightarrow Spec(k)$, and so $\\omega^N$ is the identity.\n\\end{rem}\n\n\\begin{rem}\n\t\\normalfont\n    Suppose that $I_j$ is a finite set of order $n_j$ and let $CC(Y_{\\bullet,\\bullet})$ be the bisimplicial set obtained by applying to $Y_{\\bullet,\\bullet}$ the connected components functor $CC$ that sends a connected scheme to the point and respects coproducts. Denote by $|CC(Y_{\\bullet,\\bullet})|$ its geometric realization.\n    \n    If the nonabelian cohomology $H^1(|CC(Y_{\\bullet,\\bullet})|,GL_{n_j}(R))$ is trivial, then $\\omega^N_j$ is the identity.\n    \n    In fact, $\\omega^N_j$ is an automorphism of $T^j$ in $\\DM_{eff}^-(Y_{1,\\bullet},R)$, thus an invertible element of \n    $$\\Hom_{\\DM_{eff}^-(Y_{1,\\bullet},R)}(T^j,T^j) \\cong \\prod_{I_j} \\bigoplus_{I_j}  \\Hom_{\\DM_{eff}^-(Y_{1,\\bullet},R)}(T,T)\\cong \\prod_{I_j} \\bigoplus_{I_j} H^{0,0}(Y_{1,\\bullet},R).$$ \n\tNote that $H^{0,0}(Y_{1,\\bullet},R)$ is the free $R$-module of rank given by the number of connected components of $Y_{1,\\bullet}$. Hence, if we fix a connected component of $Y_{1,\\bullet}$, $\\omega^N_j$ restricts to an invertible element of\n\t$$\\prod_{I_j} \\bigoplus_{I_j} R \\cong M_{n_j}(R),$$\n\ti.e. to an element of $GL_{n_j}(R)$.\n\tThis way one can see $\\omega^N_j$ as a morphism of groupoids\n\t$$\\Pi_1(|CC(Y_{\\bullet,\\bullet})|) \\rightarrow GL_{n_j}(R),$$\n\tso as an element of $H^1(|CC(Y_{\\bullet,\\bullet})|,GL_{n_j}(R))$.\n\\end{rem}\n\n\\begin{prop}\\label{post}\n\nLet $N$ be a motive in $\\DM_{coh}^-(Y_{\\bullet,\\bullet},R)$ such that its simplicial components $N_i$ in $\\DM_{eff}^-(Y_{i,\\bullet},R)$ are isomorphic to the direct sum $\\bigoplus_{j \\geq 0} T^j$, where $T^j$ is the possibly infinite direct sum $\\bigoplus_{I_j} T(q_j)[p_j]$ such that $(q_j)[p_j] \\prec (q_{j+1})[p_{j+1}]$ for all $j$.\n \n If $\\omega^N_j$ is trivial for any $j \\geq 0$, then there exists a Postnikov system in $\\DM_{eff}^-(Y_{\\bullet,\\bullet},R)$\n\t$$\n\t\\xymatrix{\n\t\t\\dots \\ar@{->}[r] & N^{j+1} \\ar@{->}[r] \\ar@{->}[d]\t &N^j \\ar@{->}[r] \\ar@{->}[d] &  \\dots \\ar@{->}[r]   & N^1 \\ar@{->}[r] \\ar@{->}[d]\t &N=N^0 \\ar@{->}[d] \\\\\n\t\t& T^{j+1} \\ar@{->}[ul]^{[1]}  &\tT^j \\ar@{->}[ul]^{[1]} & & T^1 \\ar@{->}[ul]^{[1]}  &\tT^0 \\ar@{->}[ul]^{[1]} & \n\t}\n\t$$\n\tsuch that the simplicial components $N^j_i$ are isomorphic to the direct sum $\\bigoplus_{k \\geq j} T^k$ and the morphisms $r_i^*(N^j \\rightarrow T^j)$ are the natural projections $\\bigoplus_{k \\geq j} T^k \\rightarrow T^j$ in $\\DM_{eff}^-(Y_{i,\\bullet},R)$.\n\\end{prop}\n\\begin{proof}\n\tTo construct the aimed Postnikov system we just need to produce morphisms $N^j \\rightarrow T^j$ where each $N^j$ is defined as the cone of the previous morphism, namely $N^j=Cone(N^{j-1} \\rightarrow T^{j-1})[-1]$. We proceed by induction.\n\t\n\tNotice that each simplicial component of $N$ is isomorphic to $\\bigoplus_{j \\geq 0} T^j$ and $T^0$ is the direct sum of possibly infinite $T(q_0)[p_0]$ such that $(q_0)[p_0] \\prec (q_j)[p_j]$ for any $j \\geq 1$ by hypothesis. By applying the triangulated functor $Lc_{\\#}$ to the filtration of Proposition \\ref{filtr}, one gets a filtration $(Lc_{\\#}N)_{\\leq n}$ for $Lc_{\\#}N$ with graded pieces $(Lc_{\\#}N)_n \\cong \\bigoplus_{j \\geq 0} \\bigoplus_{I_j} M(Y_{n,\\bullet})(q_j)[p_j+n]$. Following the lines of the proof of \\cite[Proposition 3.1.5]{smirnov.vishik}, we denote by $(Lc_{\\#}N)_{>n}$ the cone $Cone((Lc_{\\#}N)_{\\leq n} \\rightarrow Lc_{\\#}N)$ and by $(Lc_{\\#}N)_{m \\geq *>n}$ the cone $Cone((Lc_{\\#}N)_{\\leq n} \\rightarrow (Lc_{\\#}N)_{\\leq m})$ for any $m>n$. Now, note that \n\t$$(Lc_{\\#}N)_{>n} \\cong Cone(\\bigoplus_{m>n}(Lc_{\\#}N)_{m \\geq *>n} \\xrightarrow{id-sh} (Lc_{\\#}N)_{m \\geq *>n} )$$ \n\tand moreover $(Lc_{\\#}N)_{m \\geq *>n}$ is an extension of $(Lc_{\\#}N)_k$ for $n<k \\leq m$. Therefore, we have that\n\t$$\\Hom_{\\DM^{-}_{eff}(k,R)}((Lc_{\\#}N)_{>0},T^0) \\cong 0,$$\n\t$$\\Hom_{\\DM^{-}_{eff}(k,R)}((Lc_{\\#}N)_{>1},T^0) \\cong 0,$$\n\t$$\\Hom_{\\DM^{-}_{eff}(k,R)}((Lc_{\\#}N)_{>1},T^0[1]) \\cong 0,$$\n\tsince $\\Hom_{\\DM^{-}_{eff}(k,R)}(M(Y_{n,\\bullet}),T(q)[p]) \\cong 0$ for any $n \\geq 0$ and any $(q)[p] \\prec (0)[0]$. We deduce from these remarks and by applying the cohomological functor $\\Hom_{\\DM^{-}_{eff}(k,R)}(-,T^0)$ to the distinguished triangle\n\t$$(Lc_{\\#}N)_0 \\rightarrow Lc_{\\#}N \\rightarrow (Lc_{\\#}N)_{>0} \\rightarrow (Lc_{\\#}N)_0[1]$$\n\tthat there exists an exact sequence\n\t$$0 \\rightarrow \\Hom_{\\DM^{-}_{eff}(k,R)}(Lc_{\\#}(N),T^0) \\rightarrow \\Hom_{\\DM^{-}_{eff}(k,R)}(Lc_{\\#,0}(N_0),T^0)$$\n\t$$\\rightarrow \\Hom_{\\DM^{-}_{eff}(k,R)}(Lc_{\\#,1}(N_1),T^0).$$\n\tRepeating the same arguments for $T^0$ in $\\DM_{coh}^-(Y_{\\bullet,\\bullet},R)$ one gets a similar sequence \n\t$$0 \\rightarrow \\Hom_{\\DM^{-}_{eff}(k,R)}(Lc_{\\#}(T^0),T^0) \\rightarrow \\Hom_{\\DM^{-}_{eff}(k,R)}(Lc_{\\#,0}(T^0),T^0)$$\n\t$$\\rightarrow \\Hom_{\\DM^{-}_{eff}(k,R)}(Lc_{\\#,1}(T^0),T^0).$$\n\tIn order to produce an isomorphism between $\\Hom_{\\DM^{-}_{eff}(k,R)}(Lc_{\\#}(N),T^0)$ and $\\Hom_{\\DM^{-}_{eff}(k,R)}(Lc_{\\#}(T^0),T^0)$ we need to identify the last morphisms of the two exact sequences. Since in the exact sequences only the $0$th and the $1$st simplicial components appear, it is enough to get a compatibility between the coherent system $(N_i,N_{\\theta})$ and the one of $T^0$ for $i=0,1$ and simplicial maps $\\theta:[0] \\rightarrow [1]$, where $N_{\\theta}$ is the structural isomorphism $LY^*_{\\theta}(N_0) \\rightarrow N_1$ in $\\DM_{eff}^-(Y_{1,\\bullet},R)$. In other words, we want a commutative diagram\n\t$$\n\t\\xymatrix{\n\t\tN_1 \\cong \\bigoplus_{j \\geq 0} T^j \\ar@{->}[r]^{\\omega^N} \\ar@{->}[d] & N_1 \\cong \\bigoplus_{j \\geq 0} T^j \\ar@{->}[d]\\\\\n\t\tT^0  \\ar@{->}[r]_{id} & T^0.\n\t}\n\t$$\n Indeed, we have such a commutative diagram since by hypothesis $\\omega^N_0$ is trivial. Hence, the two exact sequences above coincide. Then, we have that\n$$\\Hom_{\\DM^{-}_{eff}(Y_{\\bullet,\\bullet},R)}(N,T^0) \\cong \\Hom_{\\DM^{-}_{eff}(k,R)}(Lc_{\\#}(N),T^0) \\cong$$\n$$\\Hom_{\\DM^{-}_{eff}(k,R)}(Lc_{\\#}(T^0),T^0) \\cong \\Hom_{\\DM^{-}_{eff}(Y_{\\bullet,\\bullet},R)}(T^0,T^0)$$\nby adjunctions and the identity of $T^0$ provides the pursued morphism $N \\rightarrow T^0$ whose restriction on each simplicial component is given by the natural projection $\\bigoplus_{k \\geq 0}T^k \\rightarrow T^0$. It follows that $N^1_i$ is isomorphic to $\\bigoplus_{k \\geq 1}T^k$ for any $i$. This proves the induction basis.\n\t\nNow, suppose we have a morphism from $N^k$ to $T^k$ for any $0 \\leq k \\leq j-1$, where each $N^k$ is defined as $Cone(N^{k-1} \\rightarrow T^{k-1})[-1]$. We denote by $N^j$ the cone $Cone(N^{j-1} \\rightarrow T^{j-1})[-1]$. Notice that the simplicial components of $N^j$ are all isomorphic to $\\bigoplus_{l \\geq j} T^l$ and $T^j$ is the direct sum of possibly infinite $T(q_j)[p_j]$ such that $(q_j)[p_j] \\prec (q_l)[p_l]$ for any $l \\geq j+1$ by hypothesis. Therefore, by applying the same arguments of the induction basis to $N^j$, using the fact that $\\omega^{N^j}_0=\\omega^N_j$ is trivial by hypothesis, there exists a morphism $N^j \\rightarrow T^j$. This completes the proof.\n\\end{proof}\n\n\\begin{rem}\n\t\\normalfont\n\tWe point out that, in the case it exists an integer $k$ such that $I_j$ is empty for all $j \\geq k$, the previous result provides a finite Postnikov system for $N$ in $\\DM_{eff}^-(Y_{\\bullet,\\bullet},R)$.\n\\end{rem}\n\nWe want to apply Proposition \\ref{post} to produce a spectral sequence for morphisms having motivically cellular fibers, i.e. fibers whose motives are direct sums of Tate motives satisfying certain conditions. First, we need to construct suitable Postnikov systems. The next result is a generalisation of \\cite[Proposition 4.2]{tanania.b}.\n\n\\begin{prop} \\label{Serre}\n\tLet $\\pi:X_{\\bullet,\\bullet} \\rightarrow Y_{\\bullet,\\bullet}$ be a smooth coherent morphism of smooth bisimplicial schemes over $k$, $A_{\\bullet}$ a smooth simplicial $k$-scheme and, for any $j \\geq 0$, $T^j$ the possibly infinite direct sum of Tate motives $\\bigoplus_{I_j} T(q_j)[p_j]$ in $\\DM_{eff}^-(k,R)$ such that $(q_j)[p_j] \\prec (q_{j+1})[p_{j+1}]$. Moreover, suppose the following conditions hold:\\\\\n\t1) over the $0$th simplicial component $\\pi$ is isomorphic to the projection $Y_{0,\\bullet} \\times A_{\\bullet} \\rightarrow Y_{0,\\bullet}$;\\\\\n\t2) $\\omega^{M(X_{\\bullet,\\bullet} \\rightarrow Y_{\\bullet,\\bullet})}_j$ is trivial for any $j \\geq 0$;\\\\\n\t3) $M(A_{\\bullet}) \\cong \\bigoplus_{j \\geq 0} T^j \\in \\DM_{eff}^-(k,R)$.\\\\\n\tThen, there exists a Postnikov system in $\\DM_{eff}^-(Y_{\\bullet,\\bullet},R)$\n\t$$\n\t\\xymatrix{\n\t\t\\dots \\ar@{->}[r] & N^{j+1} \\ar@{->}[r] \\ar@{->}[d]\t &N^j \\ar@{->}[r] \\ar@{->}[d] &  \\dots \\ar@{->}[r]   & N^2 \\ar@{->}[r] \\ar@{->}[d]\t &N^1 \\ar@{->}[r] \\ar@{->}[d]  &M(X_{\\bullet,\\bullet} \\xrightarrow{\\pi} Y_{\\bullet,\\bullet})=N^0 \\ar@{->}[d]\\\\\n\t\t& T^{j+1} \\ar@{->}[ul]^{[1]}  &\tT^j \\ar@{->}[ul]^{[1]} & & T^2 \\ar@{->}[ul]^{[1]}  &\tT^1 \\ar@{->}[ul]^{[1]} & T^0 \\ar@{->}[ul]^{[1]}\n\t}\n\t$$\n\tsuch that the simplicial components $N^j_i$ are isomorphic to the direct sum $\\bigoplus_{k \\geq j} T^k$ and the morphisms $r_i^*(N^j \\rightarrow T^j)$ are the natural projections $\\bigoplus_{k \\geq j} T^k \\rightarrow T^j$ in $\\DM_{eff}^-(Y_{i,\\bullet},R)$.\n\\end{prop}\n\\begin{proof}\n\tBy coherence of $\\pi$, we have that $\\pi_i:Y_{i,\\bullet} \\times A_{\\bullet} \\cong X_{i,\\bullet} \\rightarrow Y_{i,\\bullet}$ is the projection onto the first factor for any $i$. It follows that the coherent motive $N^0$ (see Remark \\ref{cohloc}) has simplicial components given by $N^0_i \\cong M(A_{\\bullet})$ in $\\DM_{eff}^-(Y_{i,\\bullet},R)$ for any $i$. Therefore, Proposition \\ref{post} implies the existence of the aimed Postnikov system in $\\DM_{eff}^-(Y_{\\bullet,\\bullet},R)$, and the proof is complete. \n\\end{proof} \n\nRecall from Section \\ref{spec} that, once constructed a Postnikov system in a triangulated category and considered a suitable cohomological functor, one can obtain a spectral sequence which may converge if some extra requirements are met. The following theorem just states the existence of a strongly convergent spectral sequence related to the Postnikov system of Proposition \\ref{Serre}.\n\n\\begin{thm}\\label{Serre2}\n\tLet $\\pi:X_{\\bullet,\\bullet} \\rightarrow Y_{\\bullet,\\bullet}$ be a smooth coherent morphism of smooth bisimplicial schemes over $k$ and $A_{\\bullet}$ a smooth simplicial $k$-scheme satisfying all conditions of Proposition \\ref{Serre}. Moreover, for any bidegree $(q)[p]$, suppose there is an integer $l$ such that $(q)[p] \\prec (q_l)[p_l]$. Then, there exists a strongly convergent spectral sequence\n\t$$E_1^{p,q,s}=\\prod_{I_s} H^{p-p_s,q-q_s}(Y_{\\bullet,\\bullet},R) \\Longrightarrow H^{p,q}(X_{\\bullet,\\bullet},R).$$\n\\end{thm}\n\\begin{proof}\n\tWe start by applying the construction of the exact couple associated to a Postnikov system of Section \\ref{spec} to the cohomological functor $\\Hom_{\\DM_{eff}^-(Y_{\\bullet,\\bullet},R)}(-,T(q))$, for any $q$. This way, we get a spectral sequence with $E_1$-page given by \n\t$$E_1^{p,q,s}=\\Hom_{\\DM_{eff}^-(Y_{\\bullet,\\bullet},R)}(T^s,T(q)[p]) \\cong \\prod_{I_s} H^{p-p_s,q-q_s}(Y_{\\bullet,\\bullet},R).$$\n\tThe filtration we are considering is defined by $F^m=\\ker(H^{**}(X_{\\bullet,\\bullet},R) \\rightarrow H^{**}(N^m,R))$. In order to get the strong convergence we need to check that $\\varinjlim_m H^{**}(N^m,R) \\cong 0$. Since all the $N^m$ are coherent motives, by Proposition \\ref{filtr} we have filtrations $(N^m)_{\\leq n}$ with graded pieces $(N^m)_n \\cong Lr_{n,\\#}r^*_n(N^m)[n]$. Hence, we have filtrations $(Lc_{\\#}N^m)_{\\leq n}$ with graded pieces\n\t$$(Lc_{\\#}N^m)_n \\cong \\bigoplus_{k \\geq m} \\bigoplus_{I_k} M(Y_{n,\\bullet})(q_k)[p_k+n].$$ \n\tNow, fix a bidegree $(q)[p]$, then by hypothesis there exists an integer $l$ such that $(q)[p] \\prec (q_l)[p_l]$, from which it follows that\n\t$$\\Hom_{\\DM_{eff}^-(k,R)}((Lc_{\\#}N^l)_n,T(q)[p]) \\cong 0$$\n\tfor any $n$. Therefore,\n\t$$\\Hom_{\\DM_{eff}^-(k,R)}(Lc_{\\#}(N^l),T(q)[p]) \\cong 0$$ \n\tfrom which we deduce by adjunction that $H^{p,q}(N^l,R) \\cong 0$ that implies, in particular, the triviality of $\\varinjlim_m H^{**}(N^m,R)$. Hence, by Theorem \\ref{SC} we obtain the result.\n\\end{proof}\n\nThe next result assures that the spectral sequence just constructed is functorial.\n\n\\begin{prop}\\label{Serre 3}\n\tLet $\\pi:X_{\\bullet,\\bullet} \\rightarrow Y_{\\bullet,\\bullet}$ and $\\pi':X'_{\\bullet,\\bullet} \\rightarrow Y'_{\\bullet,\\bullet}$ be smooth coherent morphisms of smooth bisimplicial schemes over $k$ and $A_{\\bullet}$ a smooth simplicial $k$-scheme that satisfies all conditions from Proposition \\ref{Serre} with respect to $\\pi'$ and such that the following square is cartesian with all morphisms smooth\n\t$$\n\t\\xymatrix{\n\t\tX_{\\bullet,\\bullet} \\ar@{->}[r]^{\\pi} \\ar@{->}[d]_{p_X} & Y_{\\bullet,\\bullet} \\ar@{->}[d]^{p_Y}\\\\\n\t\tX'_{\\bullet,\\bullet} \\ar@{->}[r]_{\\pi'} & Y'_{\\bullet,\\bullet}\n\t}\n\t$$\n\tThen, the induced square of motives in the category $\\DM_{eff}^-(Y'_{\\bullet,\\bullet},R)$ extends uniquely to a morphism of Postnikov systems where, for any $j \\geq 0$, $Lp_{Y\\#}T^j \\rightarrow T^j$ is given by $\\bigoplus_{I_j} M(p_Y)(q_j)[p_j]$.\n\\end{prop}\n\\begin{proof}\n\tWe denote by $N^j$ the objects from the Postnikov system of $\\pi$ and by $N'^j$ the ones from the Postnikov system of $\\pi'$. \n\t\n\tFirst, recall that, by Proposition \\ref{filtr}, there is a filtration of $Lc_{\\#}N^j$ with graded pieces \n\t$$(Lc_{\\#}N^j)_n \\cong \\bigoplus_{k \\geq j} \\bigoplus_{I_k} M(Y_{n,\\bullet})(q_k)[p_k+n].$$ \n\tIt follows that $\\Hom_{\\DM_{eff}^-(k,R)}((Lc_{\\#}N^j)_n,T^{j-1}[-1]) \\cong 0$ for any $n$ since, for any $k \\geq j$, we have that $(q_{j-1})[p_{j-1}] \\prec (q_k)[p_k]$ by hypothesis. Therefore, \n\t$$\\Hom_{\\DM_{eff}^-(Y'_{\\bullet,\\bullet},R)}(Lp_{Y\\#}N^j,T^{j-1}[-1]) \\cong \\Hom_{\\DM_{eff}^-(Y_{\\bullet,\\bullet},R)}(N^j,T^{j-1}[-1]) \\cong$$\n\t$$\\Hom_{\\DM_{eff}^-(k,R)}(Lc_{\\#}N^j,T^{j-1}[-1]) \\cong 0$$\n\tfrom which we deduce that there are no non-trivial morphisms from $Lp_{Y\\#}N^j $ to $T^{j-1}[-1]$. It follows that there exist unique morphisms $Lp_{Y\\#}N^j \\rightarrow N'^j$ fitting into a morphism of Postnikov systems in $\\DM_{eff}^-(Y'_{\\bullet,\\bullet},R)$\n\t$$\n\t\\xymatrix{\n\t\t\\dots \\ar@{->}[r] &Lp_{Y\\#}N^j \\ar@{->}[rr] \\ar@{->}[dd]& & Lp_{Y\\#}N^{j-1} \\ar@{->}[ld] \\ar@{->}[dd] \\ar@{->}[r] & \\dots\\\\\n\t\t& & Lp_{Y\\#}T^{j-1} \\ar@{->}[ul]^{[1]} \\ar@{->}[dd]& &\\\\\n\t\t\\dots \\ar@{->}[r] & N'^j \\ar@{->}[rr] & & N'^{j-1} \\ar@{->}[ld] \\ar@{->}[r] & \\dots\\\\\n\t\t& & T^{j-1} \\ar@{->}[ul]^{[1]}& &\n\t}\n\t$$\n\tIf we restrict our previous diagram to the $0$th simplicial component we obtain in $\\DM_{eff}^-(Y'_{0,\\bullet},R)$ the following morphism of Postnikov systems\n\t$$\n\t\\xymatrix{\n\t\t\\dots \\ar@{->}[r] &\\bigoplus_{k \\geq j}Lp_{Y_0\\#}T^k \\ar@{->}[rr] \\ar@{->}[dd]& & \\bigoplus_{k \\geq j-1}Lp_{Y_0\\#}T^k \\ar@{->}[ld] \\ar@{->}[dd] \\ar@{->}[r] & \\dots\\\\\n\t\t& & Lp_{Y_0\\#}T^{j-1} \\ar@{->}[ul]^{[1]} \\ar@{->}[dd]& &\\\\\n\t\t\\dots \\ar@{->}[r] & \\bigoplus_{k \\geq j} T^k \\ar@{->}[rr] & & \\bigoplus_{k \\geq j-1} T^k \\ar@{->}[ld] \\ar@{->}[r] & \\dots\\\\\n\t\t& & T^{j-1} \\ar@{->}[ul]^{[1]}& &\n\t}\n\t$$\n\twhere each triangle is split. By hypothesis, the morphism $Lp_{Y_0\\#}T^{j-1} \\rightarrow T^{j-1}$ in the previous diagram is basically given by $\\bigoplus_{I_{j-1}} M(p_{Y_0})(q_{j-1})[p_{j-1}]$, while the map $Lp_{Y_0\\#}N_0^{j-1} \\rightarrow N_0'^{j-1}$ is given by $\\bigoplus_{k \\geq {j-1}}\\bigoplus_{I_k} M(p_{Y_0})(q_k)[p_k]$.\n\t\n\tNow, note that the morphisms $Lp_{Y\\#}T^j \\rightarrow T^j$ and $\\bigoplus_{I_j} M(p_Y)(q_j)[p_j]$ are both in \n\t$$\\Hom_{\\DM_{eff}^-(Y'_{\\bullet,\\bullet},R)}(Lp_{Y\\#}T^j,T^j) \\cong \\Hom_{\\DM_{eff}^-(Y_{\\bullet,\\bullet},R)}(T^j,p_Y^*T^j) \\cong$$\n\t$$\\Hom_{\\DM_{eff}^-(Y_{\\bullet,\\bullet},R)}(T^j,T^j) \\cong \\prod_{I_j} \\bigoplus_{I_j} H^{0,0}(Y_{\\bullet,\\bullet},R)$$\n\tand, for the same reason, $(Lp_{Y\\#}T^j \\rightarrow T^j)=\\bigoplus_{I_j} M(p_{Y_0})(q_j)[p_j]$ is in \n\t$$\\Hom_{\\DM_{eff}^-(Y'_{0,\\bullet},R)}(Lp_{Y_0\\#}T^j,T^j) \\cong \\Hom_{\\DM_{eff}^-(Y_{0,\\bullet},R)}(T^j,p_{Y_0}^*T^j)\n\t\\cong$$\n\t$$\\Hom_{\\DM_{eff}^-(Y_{0,\\bullet},R)}(T^j,T^j) \\cong \\prod_{I_j} \\bigoplus_{I_j} H^{0,0}(Y_{0,\\bullet},R).$$\n\tRecall that $H^{0,0}(Y_{\\bullet,\\bullet},R)$ is the free $R$-module with rank equal to the number of connected components of $Y_{\\bullet,\\bullet}$ and, analogously, $H^{0,0}(Y_{0,\\bullet},R)$ is the free $R$-module with rank equal to the number of connected components of $Y_{0,\\bullet}$. Since, as in the argument at the end of \\cite[Proposition 3.4]{tanania.a}, the homomorphism\n\t$$r_0^*:H^{0,0}(Y_{\\bullet,\\bullet},R) \\rightarrow  H^{0,0}(Y_{0,\\bullet},R)$$\n\tis injective, we deduce that $Lp_{Y\\#}T^j \\rightarrow T^j$ and $\\bigoplus_{I_j} M(p_Y)(q_j)[p_j]$ are identified, which completes the proof.\n\\end{proof}\n\n\\begin{rem}\n\t\\normalfont\n\tIn particular, Proposition \\ref{Serre 3} guarantees that the Postnikov system of Proposition \\ref{Serre} is essentially unique.\n\\end{rem}\n\nNow, suppose for simplicity that $T^j \\cong T(q_j)[p_j]$ for any $j \\geq 0$ and consider the diagonal map $\\Delta: X_{\\bullet,\\bullet} \\rightarrow X_{\\bullet,\\bullet}  \\times_{Y_{\\bullet,\\bullet}} X_{\\bullet,\\bullet}$. The latter induces a morphism on motives $N \\rightarrow N \\otimes N$ in $\\DM_{eff}^-(Y_{\\bullet,\\bullet},R)$ that, in turn, extends to a morphism of Postnikov systems $N^k \\rightarrow (N \\otimes N)^k$, which on the cones looks like\n$$T^k \\rightarrow \\bigoplus_{J_k} T^i \\otimes T^j$$\nwhere $J_k$ is the set of all couples $(i,j)$ such that $(q_i+q_j)[p_i+p_j]=(q_k)[p_k]$.\nThe induced homomorphism in motivic cohomology\n$$\\bigoplus_{J_k} H^{**}(T^i,R) \\otimes H^{**}(T^j,R) \\rightarrow H^{**}(T^k,R)$$\nsends the element $1 \\otimes 1 \\in H^{p_i,q_i}(T^i,R) \\otimes H^{p_j,q_j}(T^j,R) \\cong H^{0,0}(Y_{\\bullet,\\bullet},R) \\otimes H^{0,0}(Y_{\\bullet,\\bullet},R)$ to $1 \\in H^{p_k,q_k}(T^k,R) \\cong H^{0,0}(Y_{\\bullet,\\bullet},R)$. In general, if $u$ is an element in $H^{**}(T^i,R)$ and $v$ is an element in $H^{**}(T^j,R)$, then we denote by $u \\cdot v$ the image of $u \\otimes v$ under the previous homomorphism.\n\nNote that the differential on the $E_1$-page $d_1^{p,q,s}:H^{p,q}(T^s,R) \\rightarrow H^{p+1,q}(T^{s-1},R)$ respects the $H^{**}(Y_{\\bullet,\\bullet},R)$-module structure. Hence, it is completely determined by $d_1^{2s,s,s}(1)$ that is the composition $T^{s-1} \\rightarrow N^s[1] \\rightarrow T^s[1]$. These observations imply the following Leibniz rule where, for brevity, we write only the last degree of the differential, i.e. the one related to the filtration.\n\n\\begin{prop}\\label{mult}\n\tWith the notations just introduced we have that\n\t$$d_1^k(a\\cdot b) = d_1^i(a)\\cdot b + a \\cdot d_1^j(b)$$\n\twhere $a$ and $b$ are respectively classes in $H^{**}(T^i,R)$ and $H^{**}(T^j,R)$. \n\\end{prop}\n\\begin{proof}\n\tSince the differential $d_1^k$ is a homomorphism of $H^{**}(Y_{\\bullet,\\bullet},R)$-modules, it is enough to prove the Leibniz formula only in the case $a=b=1$.\n\t\n\tFrom the morphism of Postnikov systems induced by the diagonal map we can extract a commutative diagram\n\t$$\\xymatrix\n\t{\n\t\tT^{k-1}\t\\ar@{->}[r] \\ar@{->}[d] & N^{k}[1] \\ar@{->}[r] \\ar@{->}[d] & T^{k}[1] \\ar@{->}[d]\\\\\n\t\t\\bigoplus_{J_{k-1}} T^i \\otimes T^j \\ar@{->}[r]  & (N \\otimes N)^{k}[1] \\ar@{->}[r]  & \\bigoplus_{J_k} T^i \\otimes T^j[1]}\n\t$$\n\twhere the composition of the top horizontal morphisms is $d_1^k(1)$. On the other hand, by degree reasons, for any $(i,j) \\in J_k$ the composition of the bottom horizontal morphisms restricts to\n\t$$(T^{i-1} \\otimes T^j) \\oplus (T^i \\otimes T^{j-1}) \\rightarrow T^i \\otimes T^j[1]$$\n\twhich is nothing but $d_1^i(1) \\otimes 1 + 1 \\otimes d_1^j(1)$. Thus, the result follows immediately.\n\\end{proof}\n\nWe would like to finish this section by establishing a comparison between the spectral sequence here presented and the Serre spectral sequence associated to a fiber bundle in topology. Recall that in topology for a fibre sequence\n$$F \\rightarrow E \\rightarrow B$$\nwith $\\pi_1(B)$ acting trivially on $H^*(F)$ one has a spectral sequence converging to $H^*(E)$\n$$E^{s,t}_2=H^s(B,H^t(F)) \\Longrightarrow H^*(E)$$\ncalled Serre spectral sequence (see for example \\cite[Theorem 15.27]{switzer}).\n\nAnalogously, our spectral sequence allows to reconstruct somehow the cohomology of the total bisimplicial scheme from the cohomology of the base and of the fiber, provided that the fiber is motivically cellular. Moreover, the triviality condition on the $\\omega^{M(X_{\\bullet,\\bullet} \\rightarrow Y_{\\bullet,\\bullet})}_j$ for any $j \\geq 0$ is reminiscent of the topological condition on the triviality of the action of $\\pi_1(B)$ on $H^*(F)$. On the other hand, the main difference between the two spectral sequences resides in how they are obtained. In fact, while the topological Serre spectral sequence is classically achieved by filtering the base, our spectral sequence is instead realized by filtering the fiber.\n\n\\section{The case of $BPGL_n$}\n\nIn this section we want to apply the spectral sequence of Theorem \\ref{Serre2} to approach the computation of the motivic cohomology of the Nisnevich classifying space of $PGL_n$. From now on we assume that the base field $k$ has characteristic not dividing $n$.\n\nFirst, we recall a few definitions. For a simplicial algebraic group $G_{\\bullet}$ denote by $EG_{\\bullet}$ the weakly contractible bisimplicial scheme defined by \n$$(EG_{\\bullet})_{i,h} = G_{h}^{i+1}$$\nwith standard face and degeneracy maps. Note that $EG_{\\bullet}$ has a right free $G_{\\bullet}$-action. Denote by $BG_{\\bullet}$ the bisimplicial scheme obtained as a quotient of $EG_{\\bullet}$ by this action. On each simplicial component $BG_{\\bullet}$ looks like\n$$(BG_{\\bullet})_{i,h} = G_{h}^{i}.$$\nBy abuse of notation, we denote by $EG_{\\bullet}$ and $BG_{\\bullet}$ also the diagonals of the respective bisimplicial schemes.\n\n\\begin{dfn}\nThe simplicial scheme $BG_{\\bullet}$ is called the Nisnevich classifying space of $G_{\\bullet}$ (\\cite[Example 1.11]{morel.voevodsky}).\n\\end{dfn}\n\n\\begin{rem}\n\t\\normalfont\nNote that the map of bisimplicial schemes $EG_{\\bullet} \\rightarrow BG_{\\bullet}$ is smooth coherent and over the 0th simplicial component is the projection $G_{\\bullet} \\rightarrow Spec(k)$. This is the reason why we need to work with bisimplicial models. Indeed, these maps provide a rich source of examples where it is possible to apply the spectral sequence in Theorem \\ref{Serre2}. In fact, if $G$ is a commutative algebraic group, then for any $n \\geq 1$ there are simplicial commutative algebraic groups $B^nG$ each of which is the (diagonal of the) classifying space of the previous one. So, we get coherent morphisms of bisimplicial schemes $EB^nG \\rightarrow B^{n+1}G$ with fiber $B^nG$. In particular, if $G=\\Gm$, then $B^nG$ is a motivic Eilenberg-MacLane space $K(\\Z,n+1,1)$, and we know from \\cite{voevodsky.em} that their motives are cellular, i.e. direct sum of Tate motives. Hence, we get Serre spectral sequences for the motivic cohomology of Eilenberg-MacLane spaces of this type.\n\\end{rem}\n\nFrom the short exact sequence of algebraic groups\n\\begin{align}\\label{cenext}\n\t1 \\rightarrow \\Gm \\rightarrow GL_n \\rightarrow PGL_n \\rightarrow 1\n\t\\end{align}\none gets the following fiber sequence\n$$B\\Gm \\rightarrow BGL_n \\rightarrow BPGL_n.$$\nFor our purposes, consider $EB\\Gm \\times BGL_n$ as a bisimplicial model for $BGL_n$ and $(EB\\Gm \\times BGL_n)\/B\\Gm$ as a bisimplicial model for $BPGL_n$ (see \\cite[Example 9.11]{jardine}). The smooth coherent morphisms of bisimplicial schemes\n$$EB\\Gm \\times BGL_n \\rightarrow  (EB\\Gm \\times BGL_n)\/B\\Gm$$\nand \n$$EB\\Gm \\rightarrow BB\\Gm$$\nare both trivial projections over the 0th simplicial component with fiber $B\\Gm$. Hence, we get a cartesian square of bisimplicial schemes\n\t\\begin{align}\\label{square}\n\\xymatrix{\n\tEB\\Gm\\times BGL_n \\ar@{->}[r] \\ar@{->}[d] & (EB\\Gm \\times BGL_n)\/B\\Gm \\ar@{->}[d]\\\\\n\tEB\\Gm \\ar@{->}[r] & BB\\Gm.\n}\n\\end{align}\n \nRecall from \\cite[Proposition 3.7]{morel.voevodsky} that $B\\Gm$ is $A^1$-homotopy equivalent to $P^{\\infty}$ whose motive is cellular. Indeed, we have that $M(P^{\\infty}) \\cong \\bigoplus_{j=0}^{\\infty} T(j)[2j]$. Therefore, we can apply the spectral sequence constructed in Theorem \\ref{Serre2} to the $BPGL_n$ case, which leads to the following result.\n\n\\begin{thm}\\label{bpg}\n\tThere exists a strongly convergent spectral sequence\n\t$$E_1^{p,q,s}=  H^{p-2s,q-s}(BPGL_n) \\Longrightarrow H^{p,q}(BGL_n)$$\n\twith differentials $d_r^{p,q,s}:E_r^{p,q,s} \\rightarrow E_r^{p+1,q,s-r}$. Moreover, the differential\n\t$$d_1^{p,q,s}:H^{p-2s,q-s}(BPGL_n) \\rightarrow H^{p-2s+3,q-s+1}(BPGL_n)$$\n\t is the multiplication by $s\\cdot d_1^{2,1,1}(1)$.\n\\end{thm}\n\\begin{proof}\n\tApplying Theorem \\ref{Serre2} to the coherent morphism $EB\\Gm \\times BGL_n \\rightarrow  (EB\\Gm \\times BGL_n)\/B\\Gm$ gives the strongly convergent spectral sequence. The description of the first differential follows easily by induction on $s$. In fact, suppose $d_1^{2s-2,s-1,s-1}(1)=(s-1) \\cdot d_1^{2,1,1}(1)$, then from Proposition \\ref{mult} we deduce that\n\t$$d_1^{2s,s,s}(1)=d_1^{2s-2,s-1,s-1}(1)+d_1^{2,1,1}(1)=s \\cdot d_1^{2,1,1}(1)$$\n\twhich concludes the proof.\n\\end{proof}\n\nSince the motivic cohomology of $BGL_n$ is known, i.e. $H^{**}(BGL_n) \\cong H^{**}(k)[c_1,\\dots,c_n]$, we can ``reverse-engineer\" the previous spectral sequence in order to obtain information about the motivic cohomology of $BPGL_n$.\n\nBefore proceeding, recall that the Chern class $c_i$ is in bidegree $(i)[2i]$ for any $i$, so $H^{p,q}(BGL_n) \\cong 0$ for $p > 2q$.\n\n\\begin{cor}\\label{triv}\n\tFor all $p \\geq 3q+1$ we have that $H^{p.q}(BPGL_n) \\cong 0$.\n\\end{cor}\n\\begin{proof}\n\tWe proceed by induction on $q$. For $q=0$, it follows from an easy inspection of the spectral sequence that $H^{p,0}(BPGL_n) \\cong H^{p,0}(BGL_n) \\cong 0$ for all $p \\geq 1$, which provides the induction basis.\n\t\n\tNow, suppose that the statement holds for all motivic weights less than $q$. The $E_1$-page of the spectral sequence is\n\t$$E_1^{p,q,s}=  H^{p-2s,q-s}(BPGL_n).$$\n\tConsider $p \\geq 3q+1$, then $p-2s \\geq 3q+1-2s \\geq 3(q-s) +1$, Hence, by induction hypothesis $E_1^{p,q,s} \\cong 0$ for all $s \\geq 1$. It follows that the only piece of the spectral sequence that contributes for $p \\geq 3q+1$ to $H^{p,q}(BGL_n) \\cong 0$ comes from $E_1^{p,q,0}=  H^{p,q}(BPGL_n)$. But the differentials $d_r: E_r^{p-1,q,r} \\rightarrow E_r^{p,q,0}$ are all trivial since $E_1^{p-1,q,r} \\cong H^{p-1-2r,q-r}(BPGL_n) \\cong 0$ as $p-1-2r \\geq 3q-2r \\geq 3(q-r)+1$.\n\t\n\tTherefore, \n\t$$H^{p,q}(BGL_n) \\cong E_{\\infty}^{p,q,0} \\cong H^{p,q}(BPGL_n)$$\n\tfor $p \\geq 3q+1$ that concludes the proof.\n\t\\end{proof}\n\nRecall from \\cite[Example 9.11]{jardine} that $BB\\Gm$ is the Eilenberg-MacLane space $K(\\Gm,2)$, so by adjunction there is a canonical element $\\chi$ in $H^{3,1}(BB\\Gm) \\cong H_{Nis}^2(BB\\Gm,\\Gm) \\cong [BB\\Gm,BB\\Gm]$ corresponding to the identity $BB\\Gm \\rightarrow BB\\Gm$ (here by $[-,-]$ we mean hom-sets in ${\\mathcal H}_s(k)$).\n\n\\begin{lem}\\label{bb}\n\tWe have that\n\t$$H^{3,1}(BB\\Gm) \\cong \\Z$$\n\tgenerated by $\\chi$.\n\\end{lem}\n\\begin{proof}\n\tWe can apply Theorem \\ref{Serre2} to the coherent morphism $EB\\Gm \\rightarrow BB\\Gm$ with fiber $B\\Gm$. Note that for $p=q=0$ the differentials are all trivial and we get\n\t$$H^{0,0}(EB\\Gm) \\cong E_{\\infty}^{0,0,0} \\cong H^{0,0}(BB\\Gm)$$\n\tfrom which it follows that $H^{0,0}(BB\\Gm) \\cong \\Z$.\n\t\n\tIn order to compute $H^{3,1}(BB\\Gm)$, the part of the $E_1$-page we need consists of the groups $E_1^{3,1,0} \\cong H^{3,1}(BB\\Gm)$ and $E_1^{2,1,1} \\cong H^{0,0}(BB\\Gm) \\cong \\Z$ linked by the differential $d_1^{2,1,1}:H^{0,0}(BB\\Gm) \\cong \\Z \\rightarrow H^{3,1}(BB\\Gm)$. Hence, we obtain \n\t$$0\\cong H^{3,1}(EB\\Gm) \\cong E_{\\infty}^{3,1,0} \\cong H^{3,1}(BB\\Gm)\/\\Ima(d_1^{2,1,1}),$$\n\t$$E_{\\infty}^{2,1,0} \\cong H^{2,1}(BB\\Gm)$$\n\tand\n\t$$E_{\\infty}^{2,1,1} \\cong  \\ker(d_1^{2,1,1}).$$\n\tTherefore, from the short exact sequence\n\t$$0 \\rightarrow E_{\\infty}^{2,1,0} \\rightarrow H^{2,1}(EB\\Gm) \\cong 0 \\rightarrow E_{\\infty}^{2,1,1} \\rightarrow 0$$\n\tone gets that $d_1^{2,1,1}$ is an isomorphism, which completes the proof.\n\t\\end{proof}\n\nThe right vertical map in \\ref{square} induces in ${\\mathcal H}_s(k)$ a class of $[BPGL_n,BB\\Gm] \\cong H^{3,1}(BPGL_n)$ that classifies the central extension \\ref{cenext} (see \\cite[Theorem 1.2]{rolle}). Denote by $x$ this canonical element. Note that $x$ is nothing but the image of $\\chi$ under the induced homomorphism $H^{3,1}(BB\\Gm) \\rightarrow H^{3,1}(BPGL_n)$.\n\n\\begin{thm}\\label{comp}\n\tIn motivic weights 0, 1 and 2 the following isomorphisms hold\n\t\\begin{align*}\n\t\tH^{p,0}(BPGL_n) &\\cong \n\t\t\\begin{cases}\n\t\t\t\\Z & p=0\\\\\n\t\t\t0 & otherwise\n\t\t\t\\end{cases}\\\\\n\t\tH^{p,1}(BPGL_n) &\\cong \n\t\t\\begin{cases}\n\t\t\tk^* & p=1\\\\\n\t\t\t\\Z\/n \\cdot x & p=3\\\\\n\t\t\t0 & otherwise\n\t\t\\end{cases}\\\\\n\tH^{p,2}(BPGL_n) &\\cong \n\t\\begin{cases}\n\t\tH^{p,2}(k) & p\\leq2\\\\\n\t\t\\mu_n(k) & p=3\\\\\n\t\tk^*\/n \\cdot x \\oplus \\Z & p=4\\\\\n\t\t\\Z\/2 \\cdot x^2 & p=6 \\: and \\: n \\: even\\\\\n\t\t0 & otherwise.\n\t\\end{cases}\n\t\t\\end{align*}\n\\end{thm}\n\\begin{proof}\n\tThe result follows from the spectral sequence in Theorem \\ref{bpg}. \n\t\n\tLet us start from the case $q=0$. Then, the only possibly non-trivial groups in the $E_1$-page are $E_1^{p,0,0} \\cong H^{p,0}(BPGL_n)$ for $p \\geq 0$. In this case differentials are all trivial and we get\n\t$$H^{p,0}(BGL_n) \\cong E_{\\infty}^{p,0,0} \\cong H^{p,0}(BPGL_n)$$\n\tfrom which it follows the motivic weight $0$ case.\n\t\n\tFor the case $q=1$, the non-trivial part of the $E_1$-page possibly consists of the groups $E_1^{p,1,0} \\cong H^{p,1}(BPGL_n)$ for $p \\geq 1$ and $E_1^{2,1,1} \\cong H^{0,0}(BPGL_n) \\cong \\Z$. There is only one non-zero differential $d_1^{2,1,1}:H^{0,0}(BPGL_n) \\cong \\Z \\rightarrow H^{3,1}(BPGL_n)$. Hence, we obtain \n\t$$H^{p,1}(BGL_n) \\cong E_{\\infty}^{p,1,0} \\cong H^{p,1}(BPGL_n)$$\n\tfor $p \\neq 2,3$,\n\t$$0\\cong H^{3,1}(BGL_n) \\cong E_{\\infty}^{3,1,0} \\cong H^{3,1}(BPGL_n)\/\\Ima(d_1^{2,1,1}),$$\n\t$$E_{\\infty}^{2,1,0} \\cong H^{2,1}(BPGL_n)$$\n\tand\n\t$$E_{\\infty}^{2,1,1} \\cong  \\ker(d_1^{2,1,1}).$$\n \tTherefore, from the short exact sequence\n \t$$0 \\rightarrow E_{\\infty}^{2,1,0} \\rightarrow H^{2,1}(BGL_n) \\rightarrow E_{\\infty}^{2,1,1} \\rightarrow 0$$\n \tone gets the exact sequence\n \t$$0 \\rightarrow H^{2,1}(BPGL_n) \\rightarrow \\Z \\rightarrow \\Z \\xrightarrow{d_1^{2,1,1}} H^{3,1}(BPGL_n) \\rightarrow 0.$$\n    At this point, we only need to understand the homomorphism in the middle $\\Z \\rightarrow \\Z$. Note that the latter is just the homomorphism $H^{2,1}(BGL_n) \\rightarrow H^{2,1}(N^1)$ induced by the Postnikov system generating the spectral sequence. Recall that $H^{2,1}(BGL_n)$ is generated by the first Chern class $c_1$ while $H^{2,1}(N^1) \\cong H^{2,1}(B\\Gm)$ is generated by the Chern class $c$. Since the map $B\\Gm \\rightarrow BGL_n$ factors through $(B\\Gm)^n$ we have that the previous homomorphism maps $c_1$ to $nc$. It follows that $H^{2,1}(BPGL_n) \\cong 0$ and $H^{3,1}(BPGL_n) \\cong \\Z\/n$ is generated by $x=d_1^{2,1,1}(1)$, by Lemma \\ref{bb} and the functoriality of the spectral sequence.\n    \n    For the case $q=2$, we have $E_1^{p,2,0} \\cong H^{p,2}(BPGL_n)$, $E_1^{3,2,1} \\cong H^{1,1}(BPGL_n) \\cong k^*$, $E_1^{5,2,1} \\cong H^{3,1}(BPGL_n) \\cong \\Z\/n$ and $E_1^{4,2,2} \\cong H^{0,0}(BPGL_n) \\cong \\Z$. The possibly non-trivial differentials on the $E_1$-page are $d_1^{4,2,2}$, $d_1^{3,2,1}$ and $d_1^{5,2,1}$. Note that, by Theorem \\ref{bpg}, $d_1^{4,2,2}$ is the multiplication by $2x$ and $d_1^{5,2,1}$ is surjective since $H^{6,2}(BGL_n)$ is trivial.\n    \n    From the short exact sequence\n    $$0 \\rightarrow E_{\\infty}^{5,2,0}\\rightarrow H^{5,2}(BGL_n) \\rightarrow E_{\\infty}^{5,2,1} \\rightarrow 0$$\n    and since $H^{5,2}(BGL_n) \\cong 0$ we get that $E_{\\infty}^{5,2,0} \\cong H^{5,2}(BPGL_n)\/ \\Ima(d_2^{4,2,2})$ and $E_{\\infty}^{5,2,1} \\cong \\ker(d_1^{5,2,1})\/ \\Ima(d_1^{4,2,2})$ are both trivial. In particular, $d_2^{4,2,2}$ is surjective and the complex\n    $$H^{0,0}(BPGL_n) \\xrightarrow{\\cdot 2x} H^{3,1}(BPGL_n) \\xrightarrow{\\cdot x} H^{6,2}(BPGL_n) \\rightarrow 0$$\n    is exact. The first homomorphism of the latter complex is $\\Z \\xrightarrow{\\cdot 2} \\Z\/n$. Hence, when $n$ is odd, it is surjective and $H^{6,2}(BPGL_n) \\cong 0$, while, when $n$ is even, its image is $\\Z\/({\\frac n 2})$ and $H^{6,2}(BPGL_n) \\cong \\Z\/2$ generated by $x^2$. \n    \n    From the short exact sequence\n    $$0 \\rightarrow E_{\\infty}^{3,2,0}\\rightarrow H^{3,2}(BGL_n) \\rightarrow E_{\\infty}^{3,2,1} \\rightarrow 0$$\n    we get the exact sequence\n    $$0 \\rightarrow H^{3,2}(BPGL_n) \\rightarrow k^* \\xrightarrow{\\cdot n} k^* \\rightarrow k^*\/n \\rightarrow 0.$$\n    Hence, $H^{3,2}(BPGL_n) \\cong \\mu_n(k)$.\n    \n     Finally, from the short exact sequence\n    $$0 \\rightarrow E_{\\infty}^{4,2,0}\\rightarrow H^{4,2}(BGL_n) \\rightarrow E_{\\infty}^{4,2,2} \\rightarrow 0$$\n    we get the exact sequence\n    $$0 \\rightarrow H^{4,2}(BPGL_n)\/\\Ima(d_2^{3,2,1}) \\rightarrow \\Z \\oplus \\Z \\rightarrow E_{\\infty}^{4,2,2} \\rightarrow 0.$$\n    Note that $E_{\\infty}^{4,2,2}$ is a subgroup of $H^{4,2}(N^2)\\cong \\Z$. The latter is generated by $c^2$ and the homomorphism $H^{4,2}(BGL_n) \\rightarrow E_{\\infty}^{4,2,2}$ maps $c_1^2$ to $n^2c^2$ and $c_2$ to ${\\frac{n(n-1)}2} c^2$. At this point we want to prove that $d_2^{4,2,2}$ is trivial. To this end, it is enough to prove that the homomorphism $H^{4,2}(BGL_n) \\rightarrow H^{4,2}(N^1)$ is surjective. First, note that since $H^{2,1}(BPGL_n) \\cong 0$ then the homomorphism $H^{4,2}(N^1) \\rightarrow H^{4,2}(N^2)$ induced by the Postnikov system is injective. Hence, $H^{4,2}(N^1) \\cong \\Z$ is generated by an element $z$ mapping to $nc^2$, if $n$ is odd, and to ${\\frac n 2}c^2$, if $n$ is even, in $H^{4,2}(N^2)$. But $c_1^2-2c_2$ in $H^{4,2}(BGL_n)$ maps to $nc^2$ in $H^{4,2}(N^2)$ if $n$ is odd, while ${\\frac n 2}c_1^2-(n+1)c_2$ maps to ${\\frac n 2}c^2$ if $n$ is even. Therefore, $d_2^{4,2,2}$ is trivial and surjective, so $H^{5,2}(BPGL_n) \\cong 0$. It immediately follows that $H^{4,2}(BPGL_n) \\cong k^*\/n \\cdot x \\oplus \\Z$ where the generator of $\\Z$ maps to $(n-1)c_1^2-2nc_2$ if $n$ is even, and to ${\\frac {n-1} 2}c_1^2-nc_2$ if $n$ is odd. This concludes the proof. \n\t\\end{proof}\n\nThe next result tells us that, as expected, the interesting part of $H^{**}(BPGL_n)$ is $n$-torsion.\n\n\\begin{prop}\n\tThere are isomorphisms of $H^{**}(k,\\Z[{\\frac 1 n}])$-algebras\n\t$$H^{**}(BPGL_n,\\Z[{\\tfrac 1 n}]) \\cong H^{**}(BSL_n,\\Z[{\\tfrac 1 n}]) \\cong  H^{**}(k,\\Z[{\\tfrac {1} {n}}])[c_2,\\dots,c_n].$$\n\\end{prop}\n\\begin{proof}\n The second isomorphism is well-known (already with $\\Z$-coefficients), so we only need to show the first one. \n \n Since the standard morphism $GL_n \\rightarrow PGL_n$ is a $\\Gm$-torsor we have a cartesian square\n \t$$\n \\xymatrix{\n \tGL_n \\times \\Gm \\ar@{->}[r]^{\\pi} \\ar@{->}[d]_{\\alpha} & GL_n \\ar@{->}[d]\\\\\n \tGL_n \\ar@{->}[r] & PGL_n\n }\n $$\n where $\\pi$ is the projection and $\\alpha$ is the $\\Gm$-action. The latter induces in turn a cartesian square\n\t$$\n\\xymatrix{\n\tSL_n \\times \\Gm \\ar@{->}[r]^{\\pi} \\ar@{->}[d]_{\\tilde{\\alpha}} & SL_n \\ar@{->}[d]\\\\\n\tGL_n \\ar@{->}[r] & PGL_n\n}\n$$\nwhere the morphism $SL_n \\rightarrow PGL_n$ (factoring through $GL_n$) is the usual $\\mu_n$-torsor.\n\nNote that $\\tilde{\\alpha}$ induces a homomorphism on the motivic cohomology of the respective classifying spaces $H^{**}(BGL_n) \\rightarrow H^{**}(BSL_n) \\otimes_{H^{**}(k)} H^{**}(B\\Gm)$ that maps $c_i$ to $c_i$ for $i \\geq 2$ and $c_1$ to $nc$. Hence, we get an isomorphism \n$$B\\tilde{\\alpha}^{*}:H^{**}(BGL_n,\\Z[{\\tfrac 1 n}]) \\cong H^{**}(BSL_n,\\Z[{\\tfrac 1 n}]) \\otimes_{H^{**}(k,\\Z[{\\frac 1 n}])}H^{**}(B\\Gm,\\Z[{\\tfrac 1 n}]).$$\n\nNow we want to prove by induction on the motivic weight that the homomorphism \n$$H^{**}(BPGL_n,\\Z[{\\tfrac 1 n}]) \\rightarrow H^{**}(BSL_n,\\Z[{\\tfrac 1 n}])$$ \nis an isomorphism. We use the functoriality of the Postnikov systems provided by Proposition \\ref{Serre 3}. For $q=0$, our spectral sequence implies that $H^{p,0}(BPGL_n,\\Z[{\\frac 1 n}]) \\cong H^{p,0}(BGL_n,\\Z[{\\frac 1 n}]) \\cong H^{p,0}(BSL_n,\\Z[{\\frac 1 n}])$ for all $p$, which provides the induction basis. Suppose that $H^{p,q'}(BPGL_n,\\Z[{\\frac 1 n}]) \\cong H^{p,q'}(BSL_n,\\Z[{\\frac 1 n}])$ for all $q' < q$ and all $p$. Since both $H^{p,q}(BPGL_n,\\Z[{\\frac 1 n}])$ and $H^{p,q}(BSL_n,\\Z[{\\frac 1 n}])$ can be reconstructed respectively from compatible extensions of $H^{p,q'}(BPGL_n,\\Z[{\\frac 1 n}])$ and $H^{p,q'}(BSL_n,\\Z[{\\frac 1 n}])$ for $q' < q$ and $H^{p,q}(BGL_n,\\Z[{\\frac 1 n}])$, five lemma implies that \n$$H^{p,q}(BPGL_n,\\Z[{\\tfrac 1 n}]) \\rightarrow H^{p,q}(BSL_n,\\Z[{\\tfrac 1 n}])$$\nis an isomorphism that is what we aimed to show.\n\t\\end{proof}\n\n\\section{The motive of a Severi-Brauer variety}\n\nThe purpose of this section is to apply previous results to obtain a description of the motive of a Severi-Brauer variety.\n\nLet $A$ be a central simple algebra of degree $n$ and $\\check C(\\seb(A))$ be the \\v{C}ech simplicial scheme of the respective Severi-Brauer variety $\\seb(A)$, i.e. $\\check C(\\seb(A))_n=\\seb(A)^{n+1}$ with face and degeneracy maps given respectively by partial projections and diagonals. Moreover, denote by $\\X_A$ the motive of $\\check C(\\seb(A))$ in $\\DM_{eff}^{-}(k)$.\n\nLet $X_A$ be the $PGL_n$-torsor associated to $A$, i.e. $X_A = {\\mathrm Iso}\\{A \\leftrightarrow M_n(k)\\}$. Note that $X_A$ is a form of $PGL_n$. Then, the scheme $(X_A \\times P^{n-1})\/PGL_n$ is a Severi-Brauer variety for $A$, i.e.\n$$\\seb(A) \\cong (X_A \\times P^{n-1})\/PGL_n.$$\n\n\\begin{prop}\\label{pssb}\n\tThere exists a Postnikov system in $\\DM_{eff}^-(k)$\n\t$$\n\t\\xymatrix{\n\t\t \\X_A(n-1)[2n-2] \\ar@{->}[r]  &M^{n-2} \\ar@{->}[r] \\ar@{->}[d] &  \\dots \\ar@{->}[r]   & M^2 \\ar@{->}[r] \\ar@{->}[d]\t &M^1 \\ar@{->}[r] \\ar@{->}[d]  &M(\\seb(A)) \\ar@{->}[d]\\\\\n\t   &\t\\X_A(n-2)[2n-4] \\ar@{->}[ul]^{[1]} & & \\X_A(2)[4] \\ar@{->}[ul]^{[1]}  &\t\\X_A(1)[2] \\ar@{->}[ul]^{[1]} & \\X_A \\ar@{->}[ul]^{[1]}\n\t}.\n\t$$\n\\end{prop}\n\\begin{proof}\n\tIt follows from \\cite[2.3.11 and Proposition 2.3.14]{smirnov.vishik} that $\\check C(\\seb(A)) \\cong (X_A \\times EPGL_n)\/PGL_n$ in ${\\mathcal H}_s(k)$. Therefore, by restricting the Postnikov system for the motive $M(BGL_n \\rightarrow BPGL_n)$ along the functor $\\DM_{eff}^-(BPGL_n) \\rightarrow \\DM_{eff}^-(\\check C(\\seb(A)))$ and then applying the forgetful functor to $\\DM_{eff}^-(k)$, one obtains a Postnikov system for the motive of $(X_A \\times EGL_n)\/GL_n$\n\t$$\n\t\\xymatrix{\n\t\t\\dots \\ar@{->}[r] & N^{j+1} \\ar@{->}[r] \\ar@{->}[d]\t &N^j \\ar@{->}[r] \\ar@{->}[d] &  \\dots \\ar@{->}[r]   \t &N^1 \\ar@{->}[r] \\ar@{->}[d]  &M((X_A \\times EGL_n)\/GL_n) \\ar@{->}[d]\\\\\n\t\t& \\X_A(j+1)[2j+2] \\ar@{->}[ul]^{[1]}  &\t\\X_A(j)[2j] \\ar@{->}[ul]^{[1]} &  &\t\\X_A(1)[2] \\ar@{->}[ul]^{[1]} & \\X_A \\ar@{->}[ul]^{[1]}\n\t}.\n\t$$\n\tNow, note that \n\t$$(X_A \\times EGL_n)\/GL_n \\cong ((X_A \\times EGL_n)\/\\Gm)\/(GL_n\/\\Gm) \\cong (X_A \\times (EGL_n\/\\Gm))\/PGL_n \\cong (X_A \\times B\\Gm)\/PGL_n.$$\n\tSince $B\\Gm$ is $A^1$-homotopy equivalent to $P^{\\infty}$, we have a map $\\seb(A) \\rightarrow (X_A \\times EGL_n)\/GL_n$ in ${\\mathcal H}(k)$ that induces a morphism on motives. From the latter we can construct a Postnikov system for $M(\\seb(A))$ that is compatible with the one for $M((X_A \\times EGL_n)\/GL_n)$. The Postnikov system for $\\seb(A)$ is actually finite since $M^j$ vanishes when restricted to a splitting field of $A$ for any $j \\geq n$.\n\t\\end{proof}\n\nLet us denote $\\ker(H^p_{\\acute{e}t}(k,\\mu_n^{\\otimes p-1}) \\rightarrow H^p_{\\acute{e}t}(k(\\seb(A)),\\mu_n^{\\otimes p-1}))$ simply by $\\ker_p$. Also, in the following results, we denote by $H^{**}_{\\acute et}(-)$ the \\'etale motivic cohomology. So, in particular, we have that \n$$H^{p,q}_{\\acute et}(-,\\Z\/n) \\cong H^p_{\\acute{e}t}(-,\\mu_n^{\\otimes q}).$$\n\n\\begin{prop}\nWe have the following isomorphisms: \n$$H^{p,q}(\\X_A) \\cong H^{p,q}(k)$$ \nfor all $p \\leq q$. Moreover,\n$$H^{p,p-1}(\\X_A) \\cong 0$$\nand \n$$H^{p+1,p-1}(\\X_A) \\cong \\ker_p$$\nfor all $p$.\n\\end{prop}\n\\begin{proof}\nBy Bloch-Kato conjecture (see \\cite[Theorems 6.16, 6.17 and 6.18]{voevodsky.motivic}), one has that $H^{p,q}(\\X_A) \\cong H^{p,q}_{\\acute{e}t}(\\X_A)$ and $H^{p,q}(k) \\cong H^{p,q}_{\\acute{e}t}(k)$ for $p \\leq q+1$. Since $H^{p,q}_{\\acute{e}t}(\\X_A) \\cong H^{p,q}_{\\acute{e}t}(k)$, we get the first two isomorphisms of the statement.\n\nRegarding the last one, again by Bloch-Kato conjecture we have that\n$$H^{p,p-1}(\\X_A,\\Z\/n) \\cong \\ker_p$$\n(see \\cite[Remark 5.3]{voevodsky.motivic}). The short exact sequence\n$$0 \\rightarrow \\Z \\xrightarrow{\\cdot n} \\Z \\rightarrow \\Z\/n \\rightarrow 0$$\ninduces a long exact sequence in motivic cohomology\n$$ \\dots \\rightarrow H^{p,p-1}(\\X_A) \\rightarrow H^{p,p-1}(\\X_A,\\Z\/n) \\rightarrow H^{p+1,p-1}(\\X_A) \\xrightarrow{\\cdot n} H^{p+1,p-1}(\\X_A) \\rightarrow \\dots.$$  \nTherefore, since $H^{p,q}(\\X_A)$ is $n$-torsion for $p \\geq q+1$ and $H^{p,p-1}(\\X_A) \\cong 0$, one obtains\n$$H^{p,p-1}(\\X_A,\\Z\/n) \\cong H^{p+1,p-1}(\\X_A)$$\nthat conludes the proof.\n\\end{proof}\n\nRecall that there is a natural homomorphism $\\alpha_A^*:H^{**}(BPGL_n) \\rightarrow H^{**}(\\X_A)$ induced by the map $\\alpha_A: (EPGL_n \\times X_A)\/PGL_n \\rightarrow BPGL_n$.\n\n\\begin{prop}\\label{xA}\n\t We have that \n\t $$\\alpha_A^*(x)=[A],$$\n\t where $x$ is the canonical class in $H^{3,1}(BPGL_n)$ from Theorem \\ref{comp} and $[A]$ is the Brauer class of $A$ in $H^{3,1}(\\X_A) \\cong \\ker_2$.\n\\end{prop}\n\\begin{proof}\nFirst note that, since $H^{3,1}(BPGL_n)$ and $H^{3,1}(\\X_A)$ are both $n$-torsion, they are respectively isomorphic to $H^{2,1}(BPGL_n,\\Z\/n)$ and $H^{2,1}(\\X_A,\\Z\/n)$. \n\nThe change of topology from Nisnevich to \\'etale gives a commutative square\n\t$$\n\\xymatrix{\n\tH^{2,1}(BPGL_n,\\Z\/n) \\ar@{->}[r] \\ar@{->}[d] & H_{\\acute{e}t}^{2,1}(BPGL_n,\\Z\/n) \\cong H^2_{\\acute{e}t}(BPGL_n,\\mu_n) \\ar@{->}[d]\\\\\n\tH^{2,1}(\\X_A,\\Z\/n) \\cong \\ker_2 \\ar@{->}[r] & H_{\\acute{e}t}^{2,1}(\\X_A,\\Z\/n) \\cong H^2_{\\acute{e}t}(k,\\mu_n)\n}\n$$\nwhere the bottom horizontal morphism is the inclusion of $\\ker_2$ in the Brauer group of $k$. By \\cite[Theorem 1.2]{rolle}, the right vertical morphism maps the central extension\n$$1 \\rightarrow \\mu_n \\rightarrow SL_n \\rightarrow PGL_n \\rightarrow 1$$\n(that is the image of $x$ under the top horizontal homomorphism: see Lemma \\ref{ce} below for more details) to the class $[A]$ in the Brauer group. Hence, we deduce that the left vertical morphism does the same, as we aimed to show.\n\\end{proof}\n\n\\begin{thm}\\label{sba}\n\tThere exists a strongly convergent spectral sequence\n\t$$E_1^{p,q,s}= \n\t\\begin{cases}\n\tH^{p-2s,q-s}(\\X_A) & 0 \\leq s \\leq n-1\\\\\n\t0 & otherwise\n\t\\end{cases}\n\t\\Longrightarrow H^{p,q}(\\seb(A))$$\n\twith differentials $d_r^{p,q,s}:E_r^{p,q,s} \\rightarrow E_r^{p+1,q,s-r}$. Moreover, the differential\n\t$$d_1^{p,q,s}:H^{p-2s,q-s}(\\X_A) \\rightarrow H^{p-2s+3,q-s+1}(\\X_A)$$\n\tis the multiplication by $s[A]$ for $1 \\leq s \\leq n-1$.\n\\end{thm}\n\\begin{proof}\nThe spectral sequence is obtained by applying motivic cohomology to the Postnikov system in Proposition \\ref{pssb}. The first differential is computed by using Theorem \\ref{bpg}, Proposition \\ref{xA} and the functoriality of the spectral sequence.\n\\end{proof}\n\n\\begin{cor}\n\tFor all $p \\geq 3q+1$ we have that $H^{p.q}(\\X_A) \\cong 0$.\n\\end{cor}\n\\begin{proof}\n\tThe proof is the same as Corollary \\ref{triv}.\n\\end{proof}\n\nAs an immediate consequence of the spectral sequence for the Severi-Brauer variety we obtain a description of the Chow group $CH^2(\\seb(A))$.\n\n\\begin{prop}\n\tThere is a short exact sequence\n\t$$0 \\rightarrow \\coker(k^* \\xrightarrow{\\cdot [A]}\\ker_3)   \\rightarrow CH^2(\\seb(A)) \\rightarrow \\Z \\rightarrow 0.$$\n\\end{prop}\n\\begin{proof}\n\tWe use the spectral sequence of Theorem \\ref{sba}. In this case the $E_1$-page is given by\n\t$$E_1^{4,2,0} \\cong H^{4,2}(\\X_A) \\cong \\ker_3,$$\n\t$$E_1^{4,2,1} \\cong H^{2,1}(\\X_A) \\cong 0,$$\n\t$$E_1^{4,2,2} \\cong H^{0,0}(\\X_A) \\cong \\Z.$$\n\tIn order to compute the $E_2$-page we also need\n\t$$E_1^{3,2,1} \\cong H^{1,1}(\\X_A) \\cong k^*,$$\n\t$$E_1^{5,2,1} \\cong H^{3,1}(\\X_A) \\cong \\ker_2.$$\n\tNote that $E_2^{4,2,2}$ is the kernel of the differential $d_1^{4,2,2}: E_1^{4,2,2} \\rightarrow E_1^{5,2,1}$, i.e. $$E_2^{4,2,2} \\cong \\ker(\\Z \\xrightarrow{\\cdot 2[A]} \\ker_2) \\cong \\Z,$$\n\twhile $E_2^{4,2,0}$ is the cokernel of $d_1^{3,2,1}:E_1^{3,2,1} \\rightarrow E_1^{4,2,0}$, i.e. the cokernel of the homomorphism $k^* \\xrightarrow{\\cdot [A]} \\ker_3$. Since $E_2^{5,2,0} \\cong E_1^{5,2,0} \\cong H^{5,2}(\\X_A)$ is $n$-torsion, we have that $E_{\\infty}^{4,2,2} \\cong E_3^{4,2,2} \\cong \\Z$. Moreover, $E_{\\infty}^{4,2,0} \\cong E_2^{4,2,0}$ and we get a filtration\n\t$$F^{4,2,0} \\hookrightarrow F^{4,2,1} \\hookrightarrow F^{4,2,2} \\cong H^{4,2}(\\seb(A))$$\n\tsuch that $E_{\\infty}^{4,2,0} \\cong F^{4,2,0}$, $E_{\\infty}^{4,2,1} \\cong F^{4,2,1}\/F^{4,2,0}$ and $E_{\\infty}^{4,2,2} \\cong F^{4,2,2}\/F^{4,2,1}$. Since $E_{\\infty}^{4,2,1} \\cong 0$, we obtain a short exact sequence\n\t$$0 \\rightarrow E_{\\infty}^{4,2,0} \\rightarrow F^{4,2,2} \\rightarrow E_{\\infty}^{4,2,2} \\rightarrow 0$$\n\tthat is exactly the one we aimed to get.\n\\end{proof}\n\nThe previous result was already obtained by Peyre in \\cite{peyre} by using different techniques. We have reported this new proof anyways as an example of a possible approach to the computation of Chow groups (and, more generally, motivic cohomology groups) of Severi-Brauer varieties by means of the spectral sequence in Theorem \\ref{sba}. Of course, in order to get any information on the torsion of $CH^i(\\seb(A))$ for $i \\geq 3$ by using our spectral sequence one should first compute $H^{p,q}(\\X_A)$ for $p \\geq q+3$, which are generally unknown, at the best of our knowledge.\n\n\\section{Torsion classes in $H^{**}(BPGL_n)$}\n\nIn this section, following \\cite{gu2} and \\cite{gu3}, we find torsion classes in the motivic cohomology of $BPGL_n$ . This allows also to generalise some results about the Chow groups of $B_{\\acute et}PGL_n$ from the complex numbers (see \\cite[Theorem 1.1]{gu2} and \\cite[Theorem 1]{gu3}) to more general fields. Indeed, we only require that the base field $k$ has characteristic not dividing $n$ and contains a primitive $n$th root of unity.\n\nFirst, let $n=p$ be an odd prime and consider the finite subgroup $C_p \\times \\mu_p$ of  $PGL_p$ described in \\cite[Section 5]{vistoli}. Recall that $C_p$ is the subgroup of the symmetric group $S_p \\subset PGL_p$ generated by the cycle $\\sigma = (1 \\: 2 \\: \\dots \\: p)$ and $\\mu_p$ is the subgroup of $PGL_p$ generated by the diagonal matrix $\\rho =[\\omega,\\dots,\\omega^{p-1},1]$, where $\\omega$ is a primitive $p$th root of unity. Note that $\\rho \\sigma= \\omega \\sigma \\rho$ in $GL_p$, so the two generators commute in $PGL_p$. The inclusion $\\iota: C_p \\times \\mu_p \\rightarrow PGL_p$ induces a homomorphism $B\\iota^*:H^{**}(BPGL_p,\\Z\/p) \\rightarrow H^{**}(B(C_p \\times \\mu_p),\\Z\/p)$. \n\nRecall from Theorem \\ref{bpg} that $H^{2,1}(BPGL_p)\\cong 0$ and $H^{3,1}(BPGL_p) \\cong \\Z\/p$, so the Bockstein homomorphism $\\bock: H^{2,1}(BPGL_p,\\Z\/p) \\rightarrow H^{3,1}(BPGL_p)$ is an isomorphism. Let $z$ be the class in $H^{2,1}(BPGL_p,\\Z\/p)$ such that $x=\\bock(z)$. By \\cite[Theorem 1.1]{rolle}, we know that $H^{2,1}_{\\acute et}(BPGL_p,\\Z\/p) \\cong H^2_{\\acute et}(BPGL_p,\\mu_p)$ is the group of central extensions of $PGL_p$ by $\\mu_p$. \n\nBefore proceeding note also that the change of topology homomorphisms\n$$H^{2,1}(-,\\Z\/p) \\rightarrow H^{2,1}_{\\acute et}(-,\\Z\/p)$$\nand \n$$H^{2,1}(-) \\rightarrow H^{2,1}_{\\acute et}(-)$$\nare respectively a monomorphism and an isomorphism for all simplicial schemes by \\cite[Theorems 6.17 and 6.18]{voevodsky.motivic}.\n\n\\begin{lem}\\label{ce}\nThe change of topology homomorphism $H^{2,1}(BPGL_p,\\Z\/p) \\rightarrow H^{2,1}_{\\acute et}(BPGL_p,\\Z\/p)$ sends $z$ to the central extension\n$$1 \\rightarrow \\mu_p \\rightarrow SL_p \\rightarrow PGL_p \\rightarrow 1.$$\n\\end{lem}\n\\begin{proof}\n\tWe have a commutative square\n\t\t$$\n\t\\xymatrix{\n\t\tH^{2,1}(BPGL_p,\\Z\/p) \\ar@{->}[r] \\ar@{->}[d]_{\\bock} & H_{\\acute{e}t}^{2,1}(BPGL_p,\\Z\/p) \\cong H^2_{\\acute et}(BPGL_p,\\mu_p) \\ar@{->}[d]^{\\bock}\\\\\n\t\tH^{3,1}(BPGL_p)  \\ar@{->}[r] & H^{3,1}_{\\acute et}(BPGL_p)\\cong H^2_{\\acute et}(BPGL_p,\\Gm) .\n\t}\n\t$$\n\tNote that $H_{\\acute{e}t}^{2,1}(BPGL_p) \\cong H^{2,1}(BPGL_p) \\cong 0$, so the Bockstein on the right is a monomorphism. Now, the statement immediately follows from the fact that $x=\\bock(z)$ maps to the central extension\n\t$$1 \\rightarrow \\Gm \\rightarrow GL_p \\rightarrow PGL_p \\rightarrow 1$$\n\tin $H^{3,1}_{\\acute et}(BPGL_p) \\cong H^2_{\\acute et}(BPGL_p,\\Gm)$.\n\t\\end{proof}\n\nIt follows from \\cite[Lemma 2.3]{rolle} that $$H^{**}(B(C_p \\times \\mu_p),\\Z\/p) \\cong H^{**}(k)[a,b,u,v]\/(a^2=b^2=0)$$\nwith $a$ and $b$ in bidegree $(0)[1]$, $u$ and $v$ in bidegree $(0)[2]$, such that $\\beta(a)=u$ and $\\beta(b)=v$ (where $\\beta$ is the reduction mod $p$ of $\\bock$).\n\n\\begin{lem}\n\tWe have that\n\t$$B\\iota^*(z)=\\lambda \\tau ab,$$\n\twhere $\\lambda $ is a non-zero element in $\\Z\/p$ and $\\tau$ is the class in $H^{0,1}(k,\\Z\/p) \\cong \\mu_p(k)$ corresponding to the primitive $p$th root of unity $\\omega$.\n\\end{lem}\t\n\\begin{proof}\n    Note that $B\\iota^*(z)$ is the class in $H^{2,1}(B(C_p \\times \\mu_p),\\Z\/p)$ that maps via the change of topology homomorphism to the central extension\n$$1 \\rightarrow \\mu_p \\rightarrow G \\rightarrow C_p \\times \\mu_p \\rightarrow 1$$\n\tinduced by the one in Lemma \\ref{ce}. The class $B\\iota^*(z)$ is non-zero since the previous extension is non-split, but restricts to a split extension both of $C_p$ and of $\\mu_p$.\n\t\n\tBy degree reasons $B\\iota^*(z)$ has the following general form\n\t$$B\\iota^*(z)=\\lambda \\tau ab +\\lambda_u \\tau u +\\lambda_v \\tau v + \\{r_a\\} a + \\{r_b\\} b$$\n\twhere $\\lambda$, $\\lambda_u$ and $\\lambda_v$ are in $\\Z\/p$ and $\\{r_a\\}$ and $\\{r_b\\}$ are in $K^M_{1}(k)\/p$. Since $B\\iota^*(z)$ restricts to zero both in $H^{2,1}(BC_p,\\Z\/p)$ and in $H^{2,1}(B\\mu_p,\\Z\/p)$, we deduce that $\\lambda_u=\\lambda_v=0$ and $\\{r_a\\}=\\{r_b\\}=0$. Therefore, $B\\iota^*(z)= \\lambda \\tau ab$ that concludes the proof.\n\t\\end{proof}\n\n\\begin{prop}\n\tThere are non-trivial classes $z_{p,k}$ in $H^{2p^{k+1}+1,p^{k+1}}(BPGL_p,\\Z\/p)$ for all $k \\geq 0$.\n\\end{prop}\n\\begin{proof}\nFor all $k \\geq 0$, define classes\n$$z_{p,k}=\\Pa^{p^k}\\Pa^{p^{k-1}} \\cdots \\Pa^p\\Pa^1\\beta(z)$$\nwhere $\\Pa^i$ are the motivic Steenrod $p$th power operations constructed in \\cite{voevodsky.reduced}.\n\nSince $\\beta(z)$ is mapped by $B\\iota^*$ to $\\lambda \\tau(ub-av)$, by Cartan formula we have that\n$$B\\iota^*(z_{p,k})= \\lambda \\tau (u^{p^{k+1}}b-av^{p^{k+1}})$$\nfor any $k\\geq 0$. Hence, $z_{p,k}$ is non-trivial for all $k$ that is what we aimed to show.\n\\end{proof}\n\n\\begin{prop}\n\tThere are non-trivial $p$-torsion classes $y_{p,k}$ in $H^{2p^{k+1}+2,p^{k+1}}(BPGL_p)$ for all $k \\geq 0$.\n\\end{prop}\n\\begin{proof}\n\tDefine $y_{p,k}$ as $\\bock(z_{p,k})$ where $\\bock:H^{**}(BPGL_p,\\Z\/p) \\rightarrow H^{**}(BPGL_p)$ is the Bockstein homomorphism. Note that the reduction mod $p$ of $y_{p,k}$ is nothing but $\\beta(z_{p,k})$ which is non-trivial since maps to $\\lambda \\tau (u^{p^{k+1}}v-uv^{p^{k+1}})$ via $B\\iota^*$. This finishes the proof.\n\t\\end{proof}\n\nNote that the classes $z$, $\\beta(z)$, $z_{p,k}$ and $\\beta(z_{p,k})$ are not $\\tau$-torsion.\n\nRecall from \\cite{morel.voevodsky} that the \\'etale classifying space $B_{\\acute et}G$ is defined as the object ${\\mathrm R}\\pi_*\\pi^*(BG)$ in ${\\mathcal H}_s(k)$, where $(\\pi^*,{\\mathrm R}\\pi_*)$ is the couple of adjoint functors induced by the morphism of sites $\\pi:(Sm\/k)_{\\acute et} \\rightarrow (Sm\/k)_{Nis}$. \n\n\\begin{prop}\n\tThere are non-trivial $p$-torsion classes ${\\upsilon}_{p,k}$ in $CH^{p^{k+1}+1}(B_{\\acute et}PGL_p)$ for all $k \\geq 0$.\n\\end{prop}\n\\begin{proof}\n\tBy \\cite[Theorem 6.17]{voevodsky.motivic} we have an isomorphism $H^{2,2}(B_{\\acute et}PGL_p,\\Z\/p) \\rightarrow H^{2,2}(BPGL_p,\\Z\/p)$. Let $\\zeta$ be the class in $H^{2,2}(B_{\\acute et}PGL_p,\\Z\/p)$ lifting $\\tau z$ and define\n\t$${\\upsilon}_{p,k}=\\bock \\Pa^{p^k}\\Pa^{p^{k-1}} \\cdots \\Pa^p\\Pa^1\\beta ({\\zeta}).$$\n\tThe classes ${\\upsilon}_{p,k}$ are non-trivial since their reductions mod $p$ map to $\\tau \\beta(z_{p,k})$.\n\t\\end{proof}\n\n\tLet $p$ be an odd prime dividing $n$. Then, the diagonal map $\\Delta: PGL_p \\rightarrow PGL_n$ induces a homomorphism $H^{**}(BPGL_n) \\rightarrow H^{**}(BPGL_p)$ that maps $x$ to $x$. Since the classes $z_{p,k}$, $y_{p,k}$ and $\\upsilon_{p,k}$ for $BPGL_p$ are constructed starting from $\\beta(z)$ (that is the reduction mod $p$ of $x$), we can define in the same way classes for $BPGL_n$. This immediately implies the following result.\n\t\n\t\\begin{cor}\\label{gentor}\n\t\tFor any odd prime $p$ dividing $n$ and $k \\geq 0$, there are non-trivial $p$-torsion classes:\n\t\t\n\t\t1) $z_{p,k}$ in $H^{2p^{k+1}+1,p^{k+1}}(BPGL_n,\\Z\/p)$;\n\t\t\n\t\t2) $y_{p,k}$ in $H^{2p^{k+1}+2,p^{k+1}}(BPGL_n)$;\n\t\t\n\t\t3) ${\\upsilon}_{p,k}$ in $CH^{p^{k+1}+1}(B_{\\acute et}PGL_n)$.\n\t\\end{cor}\n \n\n\\footnotesize{\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn this paper we consider biased positional games played on the edge set of the complete graph $K_n$ in which the winning sets are spanning subgraphs. Biased positional games were introduced by Chv\\'atal and Erd\\H{o}s~\\cite{ce1978} in 1978 and form a central part of positional games, see, for example, the monograph by Beck~\\cite{BeckBook}, or~\\cite{hkss2014, k2014} for a more recent treatment. \n\nLet $X$ be a finite set and let ${{\\mathcal F}}\\subseteq 2^X$ be a family of subsets. The set $X$ is called the {\\em board} and ${{\\mathcal F}}$ is referred to as the family of {\\em winning sets}. In the $b$-biased \\emph{Maker--Breaker} game $(X,{{\\mathcal F}})$, two players called Maker and Breaker play in rounds. In every round Maker claims one previously unclaimed element of $X$ and Breaker responds by claiming $b$ previously unclaimed elements of $X$. Maker wins if she claims  all elements of some $F\\in {{\\mathcal F}}$, otherwise Breaker wins the game. By definition a draw is impossible and thus exactly one player has a winning strategy since Maker--Breaker games are perfect information games. \n\nA certain class of games that received particular attention are Maker--Breaker games played on the edge set of the complete graph on $n$ vertices, denoted by $K_n$, in which case $X$ is the set of all unordered 2-element subsets of $K_n$, denoted by $\\binom{[n]}{2}$. In the {\\em connectivity game}, the {\\em perfect matching game}, the {\\em Hamiltonicity game}, and the {\\em triangle game}, for example, the winning sets are the edge sets of all spanning trees, all perfect matchings, all Hamilton cycles, and all copies of $K_3$, respectively. When $n$ is large enough these games are heavily in favour of Maker in the {\\em unbiased} version when $b=1$. Chv\\'atal and Erd\\H{o}s~\\cite{ce1978} therefore examined the biased variant for these games. Define the threshold bias $b^*$ of a game~$(X,{\\mathcal F})$ to be the largest integer $b$ such that Maker wins the $b$-biased Maker--Breaker game~$(X,{\\mathcal F})$. Note that Maker--Breaker games are bias-monotone, that is Maker wins for every $b\\le b^*$ and Breaker wins for every $b>b^*$. \n\nChv\\'atal and Erd\\H{o}s found that the threshold bias $b^*$ is of the order $n\\ln n$ for the connectivity, the perfect matching, and the Hamiltonicity game; and of order $\\sqrt{n}$ for the triangle game. The order of the threshold bias for an $H$-game, the game in which winning sets correspond to copies of $H$ in $K_n$, was later determined by Bednarska and {\\L}uczak~\\cite{bl2000} for any fixed graph $H$.\nExcept for the connectivity game, all the aforementioned games \ncan be cast in the following common form. Given a graph $H=H_n$,  what is the threshold bias $b^*$ of the Maker--Breaker game played on $K_n$ in which all winning sets are copies of $H_n$? In the case of Hamiltonicity we simply have $H_n = C_n$, a cycle of length $n$, and in the perfect matching game $H_n$ is a collection of $n\/2$ vertex disjoint edges. \n\nThere are choices of $H_n$ for which Maker cannot win even if $b = 1$. A trivial such example is $H_n = K_n$, however even for $H_n$ being a complete graph with only $2 \\log n$ vertices Maker cannot win the $H_n$-game \\cite[Theorem 6.4]{BeckBook}. It turns out that this can be avoided if we restrict our attention to graphs with maximum degree some constant $\\Delta$, and let $n$ be sufficiently large. Furthermore, rather than asking for the threshold bias for a specific $H_n$, we seek a \\emph{universal} upper bound: given $\\Delta$ and $n$, what is the largest $b_\\Delta = b_\\Delta(n)$ such that, on the one hand, for every graph $H_n$ with at most $n$ vertices and maximum degree at most $\\Delta$ Maker can win a $b$-biased $H_n$-game with $b \\le b_\\Delta$, and on the other hand there exists at least one such $H_n$ for which Breaker can win with bias $b = b_\\Delta$ + 1?\n\nRecently, Allen, B\\\"ottcher, Kohayakawa, Naves, and Person~\\cite{abknp2017} showed that $b_\\Delta(n)$ is of order at least $\\Omega((n\/\\log n)^{1\/\\Delta})$\\footnote{All asymptotic statements refer to $n$, the number of vertices, tending to $\\infty$.}. The triangle-preventing strategy for Breaker due to Chv\\'atal and Erd\\H{o}s~\\cite{ce1978} shows that this is tight up to a factor of $\\sqrt{\\log n}$ when $\\Delta = 2$. Furthermore, when $\\Delta = 3$ the authors of~\\cite{abknp2017} show that Breaker can win a \\emph{$K_4$-factor} game for some $b = \\Omega(n^{1\/3})$, which shows (almost) optimality in this case as well. Here the $K_4$-factor and, in general, a $K_r$-factor, corresponds to a graph $H_n$ which consists of $\\lfloor n\/r \\rfloor$ vertex-disjoint copies of $K_r$. However, the authors of~\\cite{abknp2017} have expressed a belief that their lower bound of $\\Omega((n\/\\log n)^{1\/\\Delta})$, in general, is not optimal. We provide evidence for this feeling by determining the order of the threshold bias for the $K_{\\Delta + 1}$-factor game for all $\\Delta \\ge 3$.  For $\\Delta = 3$, the threshold bias matches the upper bound in~\\cite{abknp2017}, while for $\\Delta \\ge 4$ the exponent of $n$ of the threshold bias is strictly larger than $1\/\\Delta$.\n\n\\begin{theorem} \\label{thm:main}\n  For any integer $r \\ge 4$ there exist $c, C > 0$ such that the following holds for every $n \\in r \\mathbb{Z}$.\n  \\begin{enumerate}[label={(\\roman*)}]\n    \\item\\label{main:M} If $b < c n^{2\/(r+2)}$ then Maker has a winning strategy in the $b$-biased $K_r$-factor game played on the edge set of $K_n$. \n\n    \\item\\label{main:B} If $b > C n^{2\/(r+2)}$ then Breaker has a winning strategy in the $b$-biased $K_r$-factor game played on the edge set of $K_n$. \n  \\end{enumerate}\n\\end{theorem}\n\\begin{remark}\nBy taking $c$ and $C$ to be sufficiently small and large, respectively, we have that the theorem vacuously holds for all $n < n_0$ for any chosen $n_0$. Therefore, we assume throughout the paper that $n$ is as large as needed for the calculations to be correct.\n\\end{remark}\n\nFor $b \\ge C n^{2\/(r+2)}$ we show that Breaker has a strategy to `isolate' one particular vertex from being in a copy of $K_r$, which clearly prevents Maker's graphs from containing a $K_r$-factor. Somewhat surprisingly, though not uncommon in extremal and probabilistic combinatorics, this turns out to be Breaker's best strategy: as soon as he cannot achieve this Maker is able to build a $K_r$-factor. \n\nTheorem \\ref{thm:main} suggests the following. \n\\begin{conjecture}\nFor all $\\Delta \\ge 3$, $b_\\Delta = \\Theta(n^{2\/(\\Delta + 3)}).$\n\\end{conjecture}\nIn other words, we believe that it is not significantly harder for Maker to build any other graph of maximum degree $\\Delta$ than a $K_{\\Delta+1}$-factor. \nWe take justification for this assumption from two similar settings in extremal graph theory and in random graph theory. The celebrated theorem of Hajnal and Szemer\\'edi~\\cite{hs1970} states that every graph $G$ of minimum degree at least $(1-1\/(\\Delta+1))n$ contains a $K_{\\Delta + 1}$-factor, and that condition is tight. \nBollob\\'as and Eldridge~\\cite{be1978}, and independently Catlin~\\cite{c1976}, conjectured that the condition $\\delta(G)\\ge (1-1\/(\\Delta + 1))n$ is in fact sufficient to contain every graph $H$ with $n$ vertices and maximum degree $\\Delta$. \nA similar assumption is made on the threshold bias $p^*$ for the random graph $G(n,p)$ to contain a certain graph $H_n$. Johansson, Kahn and Vu~\\cite{jkv2008} showed that $p^*(n)=(n^{-1}\\log^{1\/\\Delta}n)^{2\/(\\Delta+1)}$ is a threshold function for $G(n,p)$ to contain a $K_{\\Delta + 1}$-factor. It is folklore belief that, for every graph $H_n$ on at most $n$ vertices and of maximum degree $\\Delta$, the function  $p^*(n)$ is in fact an upper bound on the threshold functions for $G(n,p)$ to contain  $H_n$, see for example Conjecture~1.3 in~\\cite{fln2017}. Supporting evidence towards this conjecture is given by Ferber, Luh and Nguyen~\\cite{fln2017} who prove it when $H_n$ is almost-spanning, that is when $H_n$ occupies at most $(1-\\ensuremath{\\varepsilon})n$ vertices. \n\n\n\n\\medskip\n\\noindent\n{\\bf Structure of the paper.} \\\\\nIn Section 2 we take a little detour and discuss the \\emph{probabilistic intuition}, also called the {\\em Erd\\H{o}s paradigm}. While this paradigm in its basic form does not apply to the problem we consider here, a variation of it due to Allen et al.~\\cite{abknp2017} (Theorem \\ref{thm:maker_rg}) turns out to give the correct answer. This result will also serve us to provide further intuition why the threshold bias in Theorem~\\ref{thm:main} is of the order $n^{2\/(r+2)}$, or, more precisely, why Breaker is not able to isolate a single vertex from being in a copy of $K_r$ for $b < cn^{2\/(r+2)}$. In Section 3 we fix notation and state preliminary results. In Section 4, we provide Maker's strategy and prove Theorem~\\ref{thm:main}~\\ref{main:M}. Section 5 is devoted to Breaker's strategy, i.e.~Theorem~\\ref{thm:main}~\\ref{main:B}. \n\n\n\\section{Probabilistic Intuition Revised} \nChv\\'atal and Erd\\H{o}s~\\cite{ce1978} found a surprising connection between biased positional games and random graphs. Replace Maker and Breaker by RandomMaker and RandomBreaker, respectively, who choose their edges uniformly at random from all unclaimed edges. At the end of the game, the graph of RandomMaker has the same distribution as $G(n,m)$, a graph with $m$ edges chosen uniformly at random from all $\\binom{n}{2}$ possible edges, where $m$ is roughly $\\binom{n}{2}\/(b+1)$ (we omit floor and ceiling signs unless crucial). It is well known~\\cite{JLRbook} that $G(n,m)$ is (a) connected, (b) has a perfect matching, or (c) has a Hamilton cycle with probability tending to 0 if $m\\ll n \\ln n$, and with probability tending to 1 if $m\\gg n \\ln n$. That is, the threshold biases of the random version of the connectivity, the perfect matching, and the Hamiltonicity game are of the order $n\\ln n$. The results in~\\cite{ce1978} imply that the threshold bias $b^*$ in the game with clever players is of the same order of magnitude for the connectivity, the perfect matching, and the Hamiltonicity game. This phenomenon is often called the {\\em random graph intuition}, or the {\\em Erd\\H{o}s paradigm}. In fact, it turns out that the threshold biases for the random and the clever game are asymptotically equal in the connectivity game~\\cite{gs2009} and in the Hamiltonicity game~\\cite{k2011}. \n\nIt is one of the central questions in positional games to classify games for which the random graph intuition applies. A game which does very much not obey the random graph intuition is the above-mentioned triangle game or, more generally, an $H$-game for a fixed graph $H$ which contains a cycle. It is well-known that the threshold for the appearance of a triangle in $G(n,m)$ is of the order $\\Theta(n)$ (see, e.g.,~\\cite{JLRbook}). Chv\\'atal and Erd\\H{o}s~\\cite{ce1978}, however, showed that Breaker can prevent a triangle in Maker's graph when playing with a bias $b =\\Theta(\\sqrt{n})$. \n\nIt follows from Beck's winning criterion for Breaker~\\cite{b1982}, a generalisation of the classical Erd\\H{o}s-Selfridge criterion to biased games, that Breaker can {\\em always} play at least as good as RandomBreaker against RandomMaker. A result by Chv\\'atal and Erd\\H{o}s~\\cite{ce1978} shows that in some cases Breaker can play in a smarter way than just claiming edges at random. Bednarska and \\L{}uczak~\\cite{bl2000} verified that this is also the case for any $H$-game. However, the main message of their paper is not that the probabilistic intuition completely fails in these games, but rather that it has to be slightly adjusted. \n\nAs mentioned before, if both players play at random then Maker's graph is distributed as a random graph $G(n, m)$ for $m = \\binom{n}{2} \/ (b + 1)$. If Breaker does not play at random then by Maker still playing uniformly at random from the set of all \\emph{available} elements we lose control over the distribution of its graphs. To circumvent this, Bednarska and \\L{}uczak~\\cite{bl2000} suggested the following strategy for Maker: choose a next element uniformly at random from the set of \\emph{all} elements (even those that have been previously claimed) and take it only if it forms a valid move, i.e.\\ if it has not been previously claimed. Observe that Maker's graph obtained following this strategy is not a random graph but rather a subgraph obtained from a random graph after deleting a few edges. Thus, even though we might not have a fine control over the actual Maker's graph, knowing that it is a subgraph of a random graph turns out to give sufficient information to win an $H$-game. In particular, they show that when $b$ is not too large, the random graph $G(n,\\binom{n}{2}\/(b+1))$ is {\\em globally robust} with respect to containing a copy of $H$, which in turn implies that Maker has a winning strategy. That is, even after removing any small proportion of the edges the plucked random graph still contains a copy of $H$. For a precise definition of robustness we refer the reader to~\\cite{sv2008} where a systematic study of this concept was initiated. \n\nThe next step in explaining a connection between Maker--Breaker games and random graphs was done by Ferber, Krivelevich and Naves~\\cite{fkn2015}. While the strategy of playing purely at random works well in the case of $H$-games for graphs $H$ of fixed size, it fails when the winning sets are spanning subgraphs of $K_n$ as Breaker can isolate a vertex before Maker is likely to claim an edge incident to that vertex. To manifest the connection between Maker--Breaker games and random graphs for these spanning-graph games, Ferber, Krivelevich and Naves~\\cite{fkn2015} provided a {\\em local-resilience} analogue to the theorem in~\\cite{bl2000} and showed that in a $b$-biased game played on $K_n$, Maker can claim a subgraph of $G(n,m)$ for $m=\\Theta(n^2\/b)$ such that each vertex is incident to $\\Omega(n\/b)$ Maker's edges. Thus, if Maker tries to achieve a graph property ${\\mathcal P}$ that cannot be destroyed by deleting a fixed proportion of edges at each vertex then the strategy in~\\cite{fkn2015} yields a winning strategy for Maker. In particular, lower bounds on the threshold bias for several games like the perfect matching game, the connectivity and the Hamiltonicity game could be re-established this way, though with a sub-optimal constant factor. \n\nHowever, as the reader could guess, the approach via local resilience does not work for all spanning-structure Maker--Breaker games on $K_n$. The property of containing a $K_3$-factor, for example, is not locally resilient as all triangles in $G(n,m)$ containing a fixed vertex $v$ can be destroyed by removing a vanishing proportion of edges incident to every vertex, see e.g.~\\cite{huang2012bandwidth}. For the same reason, the property of containing a $K_r$-factor, $r\\ge 4$, is not locally resilient and the approach in~\\cite{fkn2015} is not applicable. \nCircumventing the short-coming of the resilience-type approaches, Allen, B\\\"ottcher, Kohayakawa, Naves, and Person~\\cite{abknp2017} finally show that Maker can also assume not only that its graph is a subgraph of a random graph with minimum degree of order $n\/b$, but also that the neighbourhood of each vertex has sufficiently many edges. The following theorem makes this precise. For a real $p \\in [0, 1]$ and an integer $n$, we write $\\Gamma \\sim G(n,p)$ if $\\Gamma$ is formed by starting with an empty graph on $n$ vertices and adding each possible edge with probability $p$, independently of all other edges. Furthermore, $\\Gamma\\sim G(n,p)$ satisfies a certain property ${\\mathcal P}$ {\\em asymptotically almost surely (a.a.s)} if the probability that $\\Gamma$ satisfies ${\\mathcal P}$ tends to 1 as $n\\to\\infty$.\n\n\\begin{theorem} \\label{thm:maker_rg}\n  For every $n$, $\\gamma = \\gamma(n) \\in (0, 1)$, $p \\ge 10^8 \\gamma^{-2} n^{-1\/2}$, and $b \\le 10^{-24} \\gamma^6 p^{-1}$ the following holds.  In the $b$-biased Maker-Breaker game played on $K_n$, for any fixed strategy of Breaker, if Maker draws a random graph $\\Gamma \\sim G(n,p)$ then a.a.s.~$\\Gamma$ is such that Maker can claim a spanning subgraph $G$ of $\\Gamma$ with $\\delta(G) \\ge (1 - \\gamma)np$ and $e_G(N_\\Gamma(v)) \\ge (1 - \\gamma)p^3 n^2 \/ 2$ for every $v \\in V(\\Gamma)$.\n\\end{theorem}\n\nUsing Theorem \\ref{thm:maker_rg} in combination with a sparse blow-up lemma from~\\cite{abhkp2016}, Allen et al.\\ \\cite{abknp2017} show that, for some $b =\\Omega ((n\/\\log n)^{1\/\\Delta})$, these {\\em neighbourhood properties} are enough for $G$ to contain all graphs of maximum degree $\\Delta$ on at most $n$ vertices. \n\n\nFor which $p$ can we guarantee that the neighbourhood properties given by Theorem \\ref{thm:maker_rg} guarantee that every vertex of $G$ is contained in a copy of $K_r$? \nThe neighbourhood $N_G(v)$ of a vertex $v$ in $G$ has size roughly $pn$, and the graph induced on $N_G(v)$ is a subgraph of $G(n,p)$ that still contains about $(1-\\ensuremath{\\varepsilon})n^2p^3\/2$ edges, i.e.~all but a small proportion of edges of $G(n,p)$ in $N_G(v)$ are also edges of $G$. That is, the subgraph of $G$ induced by $N_G(v)$ has roughly the same distribution as the random graph $G(np,p)$, and for the latter to robustly contain a copy of $K_{r-1}$ it is enough to have $p > C(np)^{-2\/r}$ for some constant $C$, which translates to $p>Cn^{-2\/(r+2)}$. It turns out that this is the main obstacle for Maker to create a $K_r$-factor.\n\n\n\n\\section{Preliminaries}\n\nWe use standard graph-theoretic notation. All considered graphs are finite and simple. Given a graph $G$, we let $e(G)$ and $v(G)$ denote its number of edges and vertices, respectively. Given a set $X \\subseteq V(G)$, let $e_G(X)$ denote the number of edges of $G$ with both endpoints in $X$. Similarly, for disjoint subsets $X, Y \\subseteq V(G)$ we let $e_G(X, Y)$ denote the number of edges of $G$ with one endpoint in $X$ and the other in $Y$. Given a vertex $v \\in V(G)$, we let $N_G(v)$ denote its neighbourhood, and for a set $X$ let $N_G(X) = \\bigcup_{v \\in X} N_G(v)$. When $G$ is clear from the context, we omit the subscript. For brevity we also omit floors and ceilings, keeping in mind that all the calculations leave enough margin to accumulate all the rounding errors. \nAll asymptotic statements refer to $n$, the number of vertices, tending to $\\infty$. \nFollowing standard asymptotic notation we write in particular, $f\\ll g$ when $f\/g \\to 0$ as $n\\to\\infty$, and $f\\gg g$ if $g\\ll f$. \n\n\\subsection{Properties of random graphs}\n\nThe following well-known estimates on the likely discrepancy of edges and the concentration of degrees in random graphs follow immediately from Chernoff's inequality and the union bound. \n\\begin{lemma} \\label{lemma:disc}\nLet $p = p(n)$ be such that $n^{-1} \\le p \\le 0.99$. Then a.a.s.~$\\Gamma \\sim\\ensuremath{G(n,p)}$ satisfies the following properties: \n  \\begin{itemize}\n    \\item For all disjoint subsets $X, Y \\subseteq V(\\Gamma)$ such that $|X| \\le |Y|$ we have\n    $$\n      e(X, Y) = |X||Y|p \\pm O\\left( |Y| \\sqrt{|X| p \\log (n\/|Y|)} \\right);\n    $$\n\n    \\item For every subset $X \\subseteq V(\\Gamma)$ we have\n    $$\n      e(X) = |X|^2p\/2 \\pm O\\left( |X|\\sqrt{|X|p \\log (n\/|X|)} \\right);\n    $$\n\n    \\item For every vertex $v \\in V(\\Gamma)$ we have \n    $$\n      |N(v)| = np \\pm O(\\sqrt{np \\log n})\n    $$\n  \\end{itemize}\n\\end{lemma}\n\nIn order to state the second result we first need some preparation. Given a graph $G$ and $\\ensuremath{\\varepsilon} \\in [0,1]$, we say that a pair of disjoint subsets $V_1, V_2 \\subseteq V(G)$ forms an \\emph{$(\\ensuremath{\\varepsilon})$-regular pair} if for $i=1,2$ and for every $V_i' \\subseteq V_i$ of size $|V_i'| \\ge \\ensuremath{\\varepsilon} |V_i|$ we have\n$$\n  \\left| e(V_1', V_2') - |V_1'||V_2'|p \\right| \\le \\ensuremath{\\varepsilon} |V_1'||V_2'| p,\n$$\nwhere $p = e(V_1, V_2)\/|V_1||V_2|$. \nNote that Lemma \\ref{lemma:disc} implies that a.a.s.~every pair of subsets of $\\ensuremath{G(n,p)}$ of size, say, at least $\\log n \/ p$, forms an $(\\ensuremath{\\varepsilon})$-regular pair for every fixed $\\ensuremath{\\varepsilon}>0$. \n\nLet $H$ be a graph with vertex set $\\{1, \\ldots, k\\}$. We denote by ${\\mathcal G}(H, n, m, \\ensuremath{\\varepsilon})$ the collection of all graphs $G$ obtained in the following way: (i) The vertex set of $G$ is a disjoint union $V_1 \\cup \\ldots \\cup V_k$ of sets of size $n$; (ii) For each edge $ij \\in E(H)$, we add to $G$ an $(\\ensuremath{\\varepsilon})$-regular bipartite graph with $m$ edges between the pair $(V_i, V_j)$. Let ${\\mathcal G}^*(H, n, m, \\ensuremath{\\varepsilon})$ denote the family of all graphs $G \\in {\\mathcal G}(H, n, m, \\ensuremath{\\varepsilon})$ which do not contain a copy of $H$. The following result, originally conjectured by Kohayakawa, \\L uczak, and R\\\"odl \\cite{kohayakawa1997onk}, was proven by Balogh, Morris, and Samotij \\cite{balogh2015independent} and, independently, Saxton and Thomason \\cite{saxton2015hypergraph}.\n\n\\begin{theorem} \\label{thm:KLR}\n  Let $H$ be a fixed graph and $\\beta > 0$. Then there exist $C, \\ensuremath{\\varepsilon} > 0$ and a positive integer $n_0$ such that \n  $$\n    \\left| {\\mathcal G}^*(H, n, m, \\ensuremath{\\varepsilon}) \\right| \\le \\beta^m \\binom{n^2}{m}^{e(H)}\n  $$\n  for every $n \\ge n_0$ and every $m \\ge Cn^{2 - 1\/m_2(H)}$, where\n  $$\n    m_2(H) = \\max\\left\\{ \\frac{e(H') - 1}{v(H') - 2} \\colon H' \\subset H, \\; v(H) \\ge 3  \\right\\}.\n  $$\n\\end{theorem}\n\nTheorem \\ref{thm:KLR} states that a random element from ${\\mathcal G}(H, n, m, \\ensuremath{\\varepsilon})$ is highly unlikely to be $H$-free. Even more, it implies that the random graph $G(n,p)$ is unlikely to contain any graph from ${\\mathcal G}^*(H, \\tilde n, m, \\ensuremath{\\varepsilon})$ for appropriate $\\tilde n$ and $p$.  This is made precise in the following lemma.\n\n\n\\begin{lemma} \\label{lemma:KLR_rg}\nLet $H$ be a graph such that $m_2(H) \\ge 2$. Then there exist $\\ensuremath{\\varepsilon}, B > 0$ such that for $n^{-1\/m_2(H)} \\le p = p(n) \\le \\ln^{-2} n$, the graph $\\Gamma \\sim \\ensuremath{G(n,p)}$ a.a.s.~has the property that, for every $\\tilde n \\ge Bp^{-m_2(H)}$, every $m \\ge \\tilde n^2 p \/ 2$ and \nevery graph $G'\\in{\\mathcal G}(H, \\tilde n, m, \\ensuremath{\\varepsilon})$, if $G' \\subseteq \\Gamma$ then $G'$ contains a copy of $H$. \n\\end{lemma}\n\n\\begin{proof}\nLet $\\ensuremath{\\varepsilon}$, $C > 0$ be as given by Theorem~\\ref{thm:KLR} for $H$ and $\\beta = (1\/(2e^2))^{e(H)}$, and set $B = (2C)^{m_2(H)}$. Let $\\Gamma\\sim\\ensuremath{G(n,p)}$. In order to prove the lemma it suffices to show that $\\mu$ vanishes, where $\\mu$ is the expected number of subgraphs of $\\Gamma$ that are isomorphic to an element in ${\\mathcal G}^*(H, \\tilde n, m, \\ensuremath{\\varepsilon})$. \nNote that  \n \\begin{align*}\n    \\mu &=  \\sum_{\\tilde n \\ge B p^{-m_2(H)}} \\sum_{m \\ge \\tilde n^2 p\/2} \\sum_{G\\in {\\mathcal G}^*} \\Pr(G\\se \\Gamma),\n\\end{align*}     \nwhere ${\\mathcal G}^* ={\\mathcal G}^*(H, \\tilde n, m, \\ensuremath{\\varepsilon})$. \nNow \n$$\n  \\Pr(G\\se \\Gamma)\\le \\binom{n}{\\tilde n k} (\\tilde n k)! p^{e(H) m}\n$$ \nwhere $k=v(H)$ for brevity. \nFurthermore, $m \\ge C \\tilde n^{2 - 1\/m_2(H)}$ follows from $m \\ge \\tilde n^2 p\/2$  and $\\tilde n \\ge B p^{-m_2(H)}$. Thus we can apply the bound on $|{\\mathcal G}^*(H, \\tilde n, m, \\ensuremath{\\varepsilon})|$ given by Theorem \\ref{thm:KLR}. \nWe therefore have that \n  \\begin{align}  \\label{aux332}  \n    \\mu &\\le \\sum_{\\tilde n \\ge B p^{-m_2(H)}} \\sum_{m \\ge \\tilde n^2 p\/2}\n    \\beta^m \\binom{\\tilde n^2}{m}^{e(H)} \\; \\binom{n}{\\tilde n k } (\\tilde n k)! p^{e(H) m} \\nonumber\\\\\n    &\\le \\sum_{\\tilde n \\ge B p^{-m_2(H)}}  \\sum_{m \\ge \\tilde n^2 p\/2} \\binom{n}{\\tilde nk} (\\tilde n k)! \n    \\beta^m \\left( \\frac{{\\tilde n}^2 e}{m}  \\right)^{e(H) m} p^{e(H) m} \\nonumber\\\\\n    &\\le \\sum_{\\tilde n \\ge B p^{-m_2(H)}} \\sum_{m \\ge \\tilde n^2 p\/2}  n^{2 \\tilde nk}\n    \\left( \\beta^{1\/e(H)} 2 e  \\right)^{e(H) m} \\nonumber\\\\\n    &\\le  \\sum_{\\tilde n \\ge Bp^{-m_2(H)}} \\sum_{m \\ge \\tilde n^2 p \/ 2}  \n    \\exp\\left(2 k\\tilde n \\ln n - m \\right),\n  \\end{align}\n  where the third inequality follows from $m\\ge \\tilde n^2p\/2$, and the last inequality follows from our choice of $\\beta$ and $e(H)\\ge 1$ since $m_2(H)\\ge 2$. Now, for sufficiently large $n$, \n  $$\n    2 k \\tilde n \\ln n - m \\le \\tilde n (2k \\ln n - \\tilde n p \/2) \\le \\tilde n (2k \\ln n - Bp^{-1}\/2) < - \\ln^2 n,\n  $$\nfrom the lower bound on $m$, the lower bound on $\\tilde n$ and $m_2(H)\\ge 2$, and from the upper bound on $p$, respectively. \nThus the final expression in~\\eqref{aux332} tends to 0 as $n\\to \\infty$. The assertion of the lemma follows from Markov's Inequality.\n\\end{proof}\n\nWe remark that the condition $m_2(H) \\ge 2$ is purely for convenience, and in fact $m_2(H) > 1$ would work as well (having an impact only on the upper bound on $p$). It should be noted that the previous lemma could also be derived from a result of Conlon, Gowers, Samotij, and Schacht \\cite{conlon2014klr}. Finally, to apply the previous result in our proof we make use of the following lemma (see, e.g., \\cite[Lemma 4.3]{gerke_steger_2005}).\n\n\n\\begin{lemma} \\label{lemma:exact_m_edges}\n  Given a positive $\\ensuremath{\\varepsilon} < 1\/6$, there exists a constant $C$ such that any $(\\ensuremath{\\varepsilon})$-regular graph $B = (V_1 \\cup V_2, E)$ contains a $(2\\ensuremath{\\varepsilon})$-regular subgraph $B = (V_1 \\cup V_2,E')$ with $|E'| = m$ edges for all $m$ satisfying $C |V(B)| \\le m \\le |E(B)|$.\n\\end{lemma}\n\n\n\n\n\\subsection{System of disjoint hyperedges}\n\nGiven a hypergraph $H$, we denote by $\\tau(H)$ the size of a smallest \\emph{vertex cover} of $H$, that is the size of a smallest subset $X \\subseteq V(H)$ that intersects all the edges of $H$. Note that if $H$ and $H'$ are hypergraphs on the same vertex set then $\\tau(H \\cup H') \\le \\tau(H) + \\tau (H')$. We make use of the following generalisation of Hall's theorem due to Haxell~\\cite{haxell1995condition}. \n\n\\begin{theorem} \\label{thm:haxell}\n  Let $H_1, \\ldots, H_t$ be a family of $r$-uniform hypergraphs on the same vertex set. If for every $I \\subseteq [t]$ we have $\\tau(\\bigcup_{i \\in I} H_i) \\ge 2 r |I|$ then one can choose a hyperedge $h_i \\in E(H_i)$ for each $i \\in [t]$ such that $h_i \\cap h_j = \\emptyset$ for distinct $i, j \\in [t]$.\n\\end{theorem}\n\nThe theorem from \\cite{haxell1995condition} gives a slightly better bound than $2 r |I|$, however for our purposes this is sufficient. \n\n\\section{Maker's strategy} \\label{sec:Maker}\n\nOur proof strategy is to show that Maker can build a graph which has certain properties and then show that these properties imply the existence of a $K_r$-factor. The properties we need are summarised in the following definition.\n\n\\begin{definition}\n  Given $\\alpha, \\beta, p \\in [0,1]$ and $r \\in \\mathbb{N}$, we say that a graph $G$ with $n$ vertices is \\emph{$(\\alpha, \\beta, p, r)$-neat} if it has the following properties:\n  \\begin{enumerate}[label={(P\\arabic*)}]\n    \\item \\label{prop:expand}\n    For every $v \\in V(G)$ we have $|N_G(v)| \\ge np\/2$ and for all disjoint $X, Y \\subseteq V(G)$ of size $|X| \\ge \\log n\/p$ and $|Y| \\ge \\alpha n$ there exists a vertex $v \\in X$ with at least $|Y|p\/2$ neighbours in $Y$; \n\n    \\item \\label{prop:in_nbr}\n    For every $v\\in V(G)$ and every subset $X \\subseteq N_G(v)$ of size $|X| \\ge \\alpha np$ the induced subgraph $G[X]$ contains a copy of $K_{r-1}$;\n\n    \\item \\label{prop:chain}\n    For all disjoint subsets $V_1, \\ldots, V_{r+1} \\subseteq V(G)$ of size $|V_i| \\ge n^{1 - \\beta}$ each, there exists a copy of $K_{r+1}^{-}$ with one vertex in each $V_i$, where $K_{r+1}^{-}$ is a graph obtained by removing an edge from a complete graph with $r+1$ vertices.\n  \\end{enumerate}\n\\end{definition}\n\nThe next lemma is the heart of the proof of Theorem \\ref{thm:main}~\\ref{main:M}. It shows that neat graphs contain $K_r$-factors under certain mild conditions on the parameters. \n\n\\begin{lemma} \\label{lemma:neat_factor}\n  For any integer $r \\ge 4$ and a positive $\\beta$ there exists positive $\\alpha$ and $n_0 \\in \\mathbb{N}$ such that any $(\\alpha, \\beta, p, r)$-neat graph with $n_0 < n \\in r \\mathbb{Z}$ vertices and $p \\ge n^{-1\/3}$ contains a $K_r$-factor.\n\\end{lemma}\n\nThe proof of Lemma \\ref{lemma:neat_factor} follows an approach from \\cite{nenadov18triangle} and we postpone it for the next subsection. We now show how Lemma \\ref{lemma:neat_factor} implies the first part of Theorem \\ref{thm:main}. \n\n\\begin{proof}[Proof of Theorem \\ref{thm:main} (i)]\nLet $\\alpha > 0$ be as given by Lemma~\\ref{lemma:neat_factor} for $\\beta = 1\/(5 (r+2)^2(r-1))$, let $K=K(\\alpha)$ be a sufficiently large integer, and suppose $n \\in r \\mathbb{Z}$ is sufficiently large. Let $p= K n^{-2\/(r+2)}$. We show that Maker can build an $(\\alpha, \\beta, p, r)$-neat graph in the $b$-biased Maker--Breaker game where $b= cp^{-1}$ for some small constant $c$. Such a graph contains a $K_r$-factor by Lemma~\\ref{lemma:neat_factor}.\n\nMaker plays two games in parallel: she plays Game 1 in every odd round and Game 2 in every even round, where Game 1 and Game 2 are defined below. Thus both games can be viewed as $(2b)$-biased Maker--Breaker games and can be played completely independently. For the rest of the argument we assume that Breaker has some fixed strategy, that is, for every disjoint pair $(E_M, E_B)$ of subsets of $E(K_n)$, which represents the current set of Maker's and Breaker's edges, he has some fixed rule what to claim next. If we can show that Maker has a winning strategy against an arbitrary such \\emph{rulebook}, then she can win regardless of what Breaker plays. In Game 1, Maker's goal is to build a graph $G_1$ that satisfies \\ref{prop:expand} and \\ref{prop:in_nbr}, and in Game 2 she builds a graph $G_2$ that satisfies \\ref{prop:chain}. Overall, this implies that $G_1 \\cup G_2$ is an $(\\alpha, \\beta, p, r)$-neat graph.\n\n\n  \\paragraph{Game 1.} \nLet $\\gamma > 0$ be a constant that we specify later, and let $\\Gamma$ be a graph on $n$ vertices that has the following properties. \n  \\begin{enumerate}[label={($\\Gamma$\\arabic*)}]\n \n    \\item \\label{game1:maker} \n    In the $(2b)$-biased Maker--Breaker game on $K_n$, Maker has a strategy to claim a spanning subgraph \n    $G \\subseteq \\Gamma$ with $\\delta(G) \\ge (1 - \\gamma)np$ and \n    $e_G(N_\\Gamma(v)) \\ge (1 - \\gamma)p^3n^2\/2$ for every $v \\in V(\\Gamma)$.\n\n    \\item \\label{game1:disc} \n    $\\Gamma$ satisfies the assertion of Lemma~\\ref{lemma:disc}.\n\n    \\item \\label{game1:r_1} \n    For every $\\tilde n\\ge Bp^{-r\/2}$, every $m\\ge \\tilde n^2 p\/2$ and every graph $G'\\in {\\mathcal G}(K_{r-1},\\tilde n, m,\\ensuremath{\\varepsilon})$, \n    if $G'\\se \\Gamma$ then $G'$ contains $K_{r-1}$ as a subgraph, \n    where $B=B(K_{r-1})$ and $\\ensuremath{\\varepsilon}=\\ensuremath{\\varepsilon}(K_{r-1})$ are the constants from Lemma~\\ref{lemma:KLR_rg} applied to $H=K_{r-1}$.\n\n\n  \\end{enumerate}\nWe argue briefly that such a graph $\\Gamma$ exists. Let $\\Gamma \\sim \\ensuremath{G(n,p)}$. \nThen $\\Gamma$ satisfies the assertion of~\\ref{game1:maker} a.a.s.~by Theorem~\\ref{thm:maker_rg} if we choose $K=K(\\gamma)$ large enough and $c=c(\\gamma)$ small enough. \nFurthermore, $\\Gamma$ satisfies \\ref{game1:disc} a.a.s.~by Lemma~\\ref{lemma:disc}, and it satisfies~\\ref{game1:r_1} by Lemma~\\ref{lemma:KLR_rg} applied to $H=K_{r-1}$ where we note that $p= K n^{-2\/(r+2)} > n^{-2\/r} = n^{-1\/m_2(K_{r-1})}$ and $m_2(K_{r-1}) \\ge 2$. Therefore, we can choose one particular graph $\\Gamma$ which has these properties. \n\nLet $G \\subseteq \\Gamma$ be a spanning subgraph guaranteed by \\ref{game1:maker}. We show that $G$ satisfies \\ref{prop:expand} and \\ref{prop:in_nbr}. \n\nFor \\ref{prop:expand} note that $G$ can be obtained from $\\Gamma$ by removing at most $2 \\gamma np$ edges touching each vertex since the maximum degree of $\\Gamma$ is at most $(1 + \\gamma)np$, by \\ref{game1:disc}, and since $\\delta(G) \\ge (1 - \\gamma)np$, by \\ref{game1:maker}. \nFurthermore, let $X, Y\\se V(G)$ be disjoint subsets of size $|X|\\ge \\log n\/p$ and $|Y|\\ge \\alpha n$, respectively. Then  $e_\\Gamma(X, Y) \\ge (1 - \\gamma)|X||Y|p$ by \\ref{game1:disc}. But then at most $|X| \\cdot 2\\gamma np$ of those edges are not present in $G$ by the preceding observation. \nBy choosing $\\gamma < \\alpha \/ 8$, we have \n  $$\n    e_G(X, Y) \\ge (1 - \\gamma)|X||Y|p - |X| \\cdot 2 \\gamma np > |X||Y|p\/2.\n  $$\n  Therefore there exists a vertex  $v \\in X$ with at least $|Y|p\/2$ neighbours in $Y$.\n\nFor \\ref{prop:in_nbr} we show that for every $v\\in V(G)$, every subset $X\\se N_G(v)$ of size $|X|\\ge \\alpha np$ hosts a copy of some $G'\\in {\\mathcal G}(K_{r-1},\\tilde n, m,\\ensuremath{\\varepsilon})$, for suitable $\\tilde n$, $m$ and $\\ensuremath{\\varepsilon}$, which  contains a copy of $K_{r-1}$ by~\\ref{game1:r_1}. Fix $v\\in V(G)$ and note that we have $|N_{\\Gamma}(v)| = (1\\pm \\gamma) np\\gg \\log n \/p$ by \\ref{game1:disc} and assumption on $p$. Thus,  again by \\ref{game1:disc},\n$$ e_\\Gamma(N_\\Gamma(v)) \\le (1 + \\gamma) |N_\\Gamma(v)|^2p\/2 \\le (1 + \\gamma)^3 n^2p^3 \/ 2 < (1 + 4 \\gamma) n^2 p^3 \/ 2, $$ \nwhere in the last inequality we assumed that $\\gamma$ is sufficiently small. From \\ref{game1:maker} we conclude that $G[N_\\Gamma(v)]$ is `missing' at most $5 \\gamma n^2 p^3 \/ 2$ edges. For brevity, let us upper bound this by $3 \\gamma n^2 p^3$. More precisely, there exists a graph $R_v$ on the vertex set $N_\\Gamma(v)$ such that $e(R_v) \\le 3 \\gamma n^2p^3$ and $G[N_\\Gamma(v)] = \\Gamma[N_\\Gamma(v)] \\setminus R_v$. Therefore, for all disjoint $X, X' \\subseteq N_\\Gamma(v)$ we have \n  $$\n     e_\\Gamma(X, X') - 3 \\gamma n^2 p^3 \\le e_G(X, X') \\le e_\\Gamma(X, X'). \n  $$\nAdditionally, if $|X|,|X'|\\gg \\log n \/p$ then $e_\\Gamma(X, X') = (1\\pm \\gamma)|X||X'|p$ by~\\ref{game1:disc}. Let $\\ensuremath{\\varepsilon}' = \\ensuremath{\\varepsilon}(K_{r-1})\/4$, where $\\ensuremath{\\varepsilon}(K_{r-1})$ is given in~\\ref{game1:r_1}. \nIt follows that for any two disjoint subsets $X,X'\\se N_{\\Gamma}(v)$ of size at least $\\ensuremath{\\varepsilon}'\\cdot (\\alpha n p\/r)$ we have \n  \\begin{align*}\n    e_G(X, X') &\\ge (1 - \\gamma)|X||X'| p - 3 \\gamma n^2 p^3  \n    \\ge \\left(1- \\ensuremath{\\varepsilon}'\\right) |X||X'| p\n  \\end{align*}\n and \n$$  e_G(X, X') \\le (1+\\gamma) |X||X'| p \\le \\left(1+\\ensuremath{\\varepsilon}'\\right) |X||X'| p,$$ \nif we choose $\\gamma$ small enough in terms of $\\ensuremath{\\varepsilon}'$, $\\alpha$ and $r$. \nThis implies that any two disjoint subsets $V_1, V_2 \\subseteq N_\\Gamma(v)$ of size $\\alpha np\/r$ form a $(2 \\ensuremath{\\varepsilon}')$-regular pair.\n\n\nLet now $X \\subseteq N_G(v) \\subseteq N_\\Gamma(v)$ be of size $\\alpha np$.  Arbitrarily choose $r-1$ disjoint subsets $V_1, \\ldots, V_{r-1} \\subseteq X$ of size $ \\tilde n = \\alpha np \/ r$. Note that  \n$ \\tilde n \\ge B p^{-r\/2} = B p^{-m_2(K_{r-1})}$, where $B=B(K_{r-1})$ is the constant given by~\\ref{game1:r_1}, since $p \\ge K n^{-2\/(r+2)}$ and $K$ is a sufficiently large constant. As previously observed, every $(V_i, V_j)$ forms a $(2\\ensuremath{\\varepsilon}')$-regular pair with $m_{i,j} = (1 \\pm \\ensuremath{\\varepsilon}') \\tilde n^2 p$ edges,\n  thus we can apply Lemma \\ref{lemma:exact_m_edges} to each pair $(V_i, V_j)$ in order to obtain a subset $E_{ij} \\subseteq E_G(V_i, V_j)$ of size exactly \n  $$\n    m = \\tilde n^2 p \/2\n  $$\n  such that $(V_i, V_j)$ is $(4\\ensuremath{\\varepsilon}')$-regular, i.e.~$(\\ensuremath{\\varepsilon})$-regular, with respect to $E_{ij}$. This gives us a graph $G' \\in \\ensuremath{\\mathcal G}(K_{r-1}, \\tilde n, m, \\ensuremath{\\varepsilon})$. As $G'$ is a subgraph of $\\Gamma$, from \\ref{game1:r_1} we conclude that it contains a copy of $K_{r-1}$. Thus, $G$ satisfies~\\ref{prop:in_nbr}.\n\n\n  \\paragraph{Game 2.} The properties \\ref{prop:in_nbr} and \\ref{prop:chain} are achieved in a fairly similar way. Thus, the analysis of Game 2 follows along the lines of the second part of Game 1. There are some crucial differences in the choice of parameters though. \nRecall that $\\beta = 1\/(5(r+2)^2(r-1))$ and that for property \\ref{prop:chain} we want to find a copy of $K_{r+1}^-$ in certain sets of size $n^{1-\\beta}$. Let $H=K_{r+1}^-$, let now $\\gamma = n^{-3\\beta}$ and let $q \\in (0,1)$ satisfy \n \\begin{equation}\\label{qBounds}\nn^{-\\frac{1-\\beta}{m_2(H)}}\\ll q \\ll n^{-\\frac{2}{r+2}} \\gamma^6.\n\\end{equation} \nNote that this is possible since $m_2(K_{r+1}^-)= \\frac{r+2}{2}-\\frac{1}{r-1}$ and by choice of $\\beta$. Delicate choices for parameters $\\gamma$ and $q$ will become apparent soon. We claim that a random graph $\\Gamma \\sim \\ensuremath{G(n,q)}$ has the following properties with high probability. \n  \\begin{enumerate}[label={($\\Gamma$\\arabic*)}]\n    \\item \\label{game2:maker} In the $(2b)$-biased Maker--Breaker game on $K_n$,  Maker has a strategy to claim a spanning subgraph $G \\subseteq \\Gamma$ with $\\delta(G) \\ge (1 - \\gamma)nq$ and $e_G(N_\\Gamma(v)) \\ge (1 - \\gamma)q^3n^2\/2$ for every $v \\in V(\\Gamma)$.\n\n    \\item \\label{game2:disc} $\\Gamma$ satisfies the assertion of Lemma \\ref{lemma:disc};\n\n    \\item \\label{game2:r_1} \n      For every $\\tilde n\\ge Bq^{-m_2(H)}$, every $m\\ge \\tilde n^2 q\/2$ and every graph $G'\\in {\\mathcal G}(H,\\tilde n, m,\\ensuremath{\\varepsilon})$, \n    if $G'\\se \\Gamma$ then $G'$ contains $H$ as a subgraph, \n    where now $B=B(H)$ and $\\ensuremath{\\varepsilon}=\\ensuremath{\\varepsilon}(H)$ are the constants from Lemma~\\ref{lemma:KLR_rg} applied to $H=K_{r+1}^-$.\n    \n  \\end{enumerate}\nLet us verify that $\\Gamma$  indeed has these properties~a.a.s. For~\\ref{game2:maker} let us verify that the conditions of \nTheorem~\\ref{thm:maker_rg} hold. Firstly, $q\\ge 10^{8}\\gamma^{-2}n^{-1\/2}$ is implied by the lower bound in~\\eqref{qBounds} since $r\\ge 4$ and since $\\beta< 1\/50$, say. Secondly, recall that $b= cp^{-1}=O(n^{2\/(r+2)})$. This together with the upper bound in~\\eqref{qBounds} implies that $\\gamma^{6} q^{-1} \\gg b$, so that \\ref{game2:maker} holds a.a.s.~by Theorem~\\ref{thm:maker_rg}.\nJust as in Game 1, $\\Gamma$ satisfies~\\ref{game2:disc}~a.a.s.~by Lemma~\\ref{lemma:disc}. \nFinally, note that the lower bound in~\\eqref{qBounds} implies in particular that $q\\ge n^{-1\/m_2(H)}$. Thus~\\ref{game2:r_1}\nholds a.a.s.~by Lemma~\\ref{lemma:KLR_rg} applied to $H=K_{r+1}^-$. \n  \nFix $\\Gamma$ with these three properties and let $G \\subseteq \\Gamma$ be a spanning subgraph satisfying \\ref{game2:maker}. Crucially, we have sacrificed the value of $q$, which is now significantly smaller than $n^{-2\/(r+2)}$, in order to get a smaller error term $\\gamma$. Note that in order to guarantee that $G$ satisfies property \\ref{prop:in_nbr} in Game 1 we needed $p = \\Omega(n^{-2\/(r+2)})$. Here it will turn out that a smaller $p$ (which we denote by $q$) suffices provided $\\gamma$ is sufficiently small. We now make this precise.\n\nFor $\\ensuremath{\\varepsilon}' = \\ensuremath{\\varepsilon}(H) \/4 >0$ consider disjoint subsets $X, X' \\subseteq V(G)$ of size at least $\\ensuremath{\\varepsilon}' n^{1 - \\beta}$. \nFirst note that \n  \\begin{align*}\n  e_{\\Gamma} (X,X') = (1\\pm \\gamma)|X||X'|q, \n  \\end{align*}\n  by~\\ref{game2:disc} since $\\gamma^2n^{1-\\beta}q\\gg \\log n$ by choice of $\\beta$ being small enough and~\\eqref{qBounds}.    \nAs before, we have that $G$ is obtained from $\\Gamma$ by removing at most $2 \\gamma nq$ edges touching each vertex, which sums to at most $\\gamma n^2 q$ removed edges in total. This, together with the above estimate on  $  e_{\\Gamma} (X,X')$, implies that\n  \\begin{align*}\n    (1 + \\gamma)|X||X'| q \\ge e_G(X, X') &\\ge (1 - \\gamma)|X||X'|q - \\gamma n^2 q \\\\\t&\\ge (1 - \\ensuremath{\\varepsilon}')|X||X'|q, \n  \\end{align*}\n  since $\\gamma < \\ensuremath{\\varepsilon}'\/2$, say, and $\\gamma n^2 = n^{2-3\\beta}\\ll \\ensuremath{\\varepsilon}' |X||X'|\/2$.  Note that it was crucial here that $\\gamma \\ll n^{- 2\\beta}$, that is, $\\gamma$ polynomially depends on $n$. Therefore, every pair of disjoint subsets $X, Y \\subseteq V(G)$ of size $n^{1 - \\beta}$ forms a $(2\\ensuremath{\\varepsilon}')$-regular pair. \n\n The rest of the argument is the same as in the previous case. Consider some disjoint $V_1, \\ldots, V_{r+1} \\subseteq V(G)$, each of size $\\tilde n = n^{1 - \\beta}$. As observed, each pair $(V_i, V_j)$ forms a $(2\\ensuremath{\\varepsilon}')$-regular pair with $m_{ij} \\ge (1 - \\ensuremath{\\varepsilon}')\\tilde n^2 q$ edges. By Lemma \\ref{lemma:exact_m_edges}, there exists a subset $E_{ij} \\subseteq E_G(V_i, V_j)$ of size exactly\n  $$\n    m = \\tilde n^2 q \/ 2\n  $$\n  such that $(V_i \\cup V_j, E_{ij})$ is $(4\\ensuremath{\\varepsilon}')$-regular, i.e.~$(\\ensuremath{\\varepsilon})$-regular. This gives us a subgraph $G' \\subseteq G$ which belongs to $\\ensuremath{\\mathcal G}(K_{r+1}^-, \\tilde n, m, \\ensuremath{\\varepsilon})$. From the lower bound in~\\eqref{qBounds} we infer that $\\tilde n \\ge B(H) q^{-m_2(H)}.$   Thus \\ref{game2:r_1} implies that $G'$ contains a copy of $K_{r+1}^-$ with one vertex in each $V_i$. \n\\end{proof}\n\n\n\n\n\\subsection{Neat graphs contain $K_r$-factors (Lemma \\ref{lemma:neat_factor})}\n\nThe proof of Lemma \\ref{lemma:neat_factor} closely follows ideas from \\cite{nenadov18triangle} which are, in turn, based on ideas of Krivelevich \\cite{krivelevich1997triangle}. Recall that $K_{r+1}^-$ denotes the graph obtained from $K_{r+1}$ by removing an edge. The main building block in the proof is an \\emph{$(r, \\ell)$-chain}, the graph obtained by sequentially `gluing' $\\ell \\ge 0$ copies of $K_{r+1}^-$ on a vertex of degree $r-1$ (see Figure \\ref{fig:chain}). We define the $(r, 0)$-chain to be a single vertex. A graph is an \\emph{$r$-chain} if it is isomorphic to an $(r, \\ell)$-chain, for some integer $\\ell \\ge 0$.\n\n\\begin{figure}[h!]   \n  \\centering\n  \\begin{tikzpicture}[scale = 0.6]\n    \\tikzstyle{blob} = [fill=black,circle,inner sep=1.7pt,minimum size=0.5pt]\n    \\tikzstyle{sq} = [fill=black,rectangle,inner sep=2.5pt,minimum size=2.5pt]\n\n    \n    \\foreach \\x in {1,...,4}{\n      \\node[sq] (f\\x) at (0 + 5 * \\x, 0) {};\n      \\foreach \\c\/\\y\/\\z in {1\/2.5\/1, 2\/2.5\/-1, 3\/2\/0.3, 4\/3\/-0.3}\n        \\node[blob] (v\\x\\c) at (0 + 5 * \\x + \\y, \\z) {};       \n    }\n    \\node[sq] (f5) at (25, 0) {};\n\n    \\foreach \\x\/\\d in {1\/1,2\/2,3\/3,4\/4}{\n      \\foreach \\c in {1,...,4}\n        \\draw (f\\x) -- (v\\x\\c);\n\n      \\draw (v\\x1) -- (v\\x2) -- (v\\x3) -- (v\\x4) -- (v\\x1);\n      \\draw (v\\x1) -- (v\\x3); \\draw (v\\x2) -- (v\\x4);\n    }\n\n    \\foreach \\x\/\\d in {2\/1,3\/2,4\/3,5\/4}{\n      \\foreach \\c in {1,...,4}\n        \\draw (f\\x) -- (v\\d\\c);      \n    }\n\n    \\node [below=2pt of f4] {$v$};\n\n    \\foreach \\x in {1,...,3}\n      \\draw[dashed, rounded corners] (-0.5 + 5 * \\x, 1.5) rectangle (0.5 + 5 * \\x + 3, -1.5);\n\n    \\draw[dashed, rounded corners] (-0.5 + 5 * 4 + 2, 1.5) rectangle (0.5 + 5 * 5, -1.5);\n  \\end{tikzpicture}\n  \\caption{The $(5, 4)$-chain with a $K_5$-factor after removing vertex $v$.} \n  \\label{fig:chain}  \n\\end{figure}\n\nAn $(r, \\ell)$-chain contains $\\ell + 1$ vertices such that removing either of them (but exactly one!) results in a graph which contains a $K_r$-factor. We call such vertices \\emph{removable}. If a graph $H$ is an $r$-chain then we use $R(H)$ to denote the set of its removable vertices. We repeatedly use the following observation.\n\n\\begin{observation} \\label{obs:canonical}\n  Let $G$ be a graph and $C_1, \\ldots, C_r \\subseteq G$ be vertex disjoint $r$-chains. If there exists a copy of $K_r$ in $G$ which intersects each $R(C_i)$ then the subgraph of $G$ induced by $\\bigcup_{i \\in [r]} V(C_i)$ contains a $K_r$-factor.\n\\end{observation}\n\nThe following lemma together with property \\ref{prop:chain} ensures the existence of large $(r, \\ell)$-chains. \n\\begin{lemma} \\label{lemma:long_chain}\nLet $G$ be a graph with $n$ vertices such that for every disjoint $X, Y \\subseteq V(G)$, each of size at least $\\alpha n$, there exists a copy of $K_{r+1}^-$ in $G$ with one vertex of degree $r-1$ in $X$ and all other vertices in $Y$. Then $G$ contains an $(r, \\ell$)-chain for every $\\ell < (1 - (r+2)\\alpha) n\/r$. \n\\end{lemma}\nIn the case when $r=3$ this is Lemma~3.1 in~\\cite{nenadov18triangle}. Trivial adjustments to that proof give Lemma~\\ref{lemma:long_chain}. We omit the proof.\n\nThe following, somewhat technical looking lemma provides a crucial \\emph{absorbing} property of a collection of $r$-chains that we exploit in the proof of Lemma \\ref{lemma:neat_factor}.\n\n\\begin{lemma} \\label{lemma:absorbing}\n  Let $G$ be a graph with $n$ vertices which satisfies \\ref{prop:chain} for some $\\beta > 0$ and $r$, where $n \\ge n_0(\\beta, r)$ is sufficiently large. Let $W \\subseteq V(G)$ be a subset of size $|W| \\ge n\/8$, and let $\\ell, t \\in \\mathbb{N}_0$ and $\\ell' \\ge \\ell, t' \\in \\mathbb{N}$ be such that:\n  \\begin{itemize}\n    \\item $(\\ell + 1) t' > n^{1 - \\beta\/2}$, and\n    \\item $(t + t')(r \\ell + 1) < |W|\/2$.\n  \\end{itemize}\n  Suppose we are given disjoint $(r, \\ell')$-chains $C_1', \\ldots, C'_{t'} \\subset V(G) \\setminus W$. Then there exist disjoint $(r, \\ell)$-chains $C_1, \\ldots, C_t \\subset G[W]$ with the following property: for every $L \\subseteq [t]$ there exists $L' \\subseteq [t']$ such that the subgraph of $G$ induced by\n  $$\n    \\left( \\bigcup_{i \\in L} V(C_i) \\right) \\cup \\left( \\bigcup_{i \\in L'} V(C'_i) \\right)\n  $$\n  contains a $K_r$-factor.\n\\end{lemma}\n\n\n\nBefore we prove Lemma \\ref{lemma:absorbing} it is instructive to first see how it is used to derive Lemma \\ref{lemma:neat_factor}. \n\n\\begin{proof}[Proof of Lemma \\ref{lemma:neat_factor}]\n  Consider an equipartition $V(G) = V_1 \\cup V_2$ chosen uniformly at random. As each vertex has $np\/2 \\gg \\log n$ neighbours (follows from \\ref{prop:expand} and the bound on $p$), by Chernoff's inequality and union-bound we have with high probability that every vertex has at least $np\/8$ neighbours in $V_1$. Therefore there exists a partition for which this holds. \n\n  Without loss of generality we may assume that $\\beta = 1\/k$ for some integer $k \\ge 2$. For each $i \\in \\{1, \\ldots, 4k-1\\}$ set $\\ell_i = n^{1 - (4k-i)\/4k}$ and $t_i = n \/ (32 k (r \\ell_i + 1))$. Note that \n  \\begin{equation} \\label{eq:ell_i_t_i}\n    (\\ell_i + 1) t_{i+1} = \\Theta(n^{1 - \\beta\/4}).\n  \\end{equation}\n\n  By repeated application of Lemma \\ref{lemma:long_chain} we can find a collection $C_1^{4k-1}, \\ldots, C_{t_{4k-1}}^{4k-1} \\subseteq G[V_2]$ of pairwise vertex-disjoint $(r, \\ell_{4k-1})$-chains. Let us elaborate briefly why this is indeed possible. Such chains occupy $t_{4k - 1} \\cdot (r \\ell_{4k - 1} + 1) < n \/ 32$ vertices. Thus if we greedily choose them one by one the set $W \\subseteq V_2$ of unoccupied vertices in $V_2$ after every step is of size at least, say, $|W| \\ge n\/4$. Therefore, by \\ref{prop:chain} we have that $G[W]$ satisfies  the assumption of Lemma \\ref{lemma:long_chain} for any constant $\\alpha > 0$, and consequently it contains an $(r, \\ell)$-chain for $\\ell < (1 - (r+2)\\alpha) |W| \/ r$. As $\\ell_{4k-1} = o(n)$, this proves our claim.\n\n  Let $U_{4k - 1} = \\bigcup_{i \\in [t_{4k - 1}]} V(C_i^{4k - 1})$. For each $i = 4k - 2, \\ldots, 1$, iteratively, let $C_1^i, \\ldots, C_{t_i}^i \\subset G[V_2] \\setminus U_{i+1}$ be disjoint $(r, \\ell_i)$-chains given by Lemma \\ref{lemma:absorbing} for $C_1^{i+1}, \\ldots, C_{t_{i+1}}^{i+1}$ (as $C_1', \\ldots, C'_{t'}$) and $W_i = V_2 \\setminus U_{i+1}$, and set $U_i = U_{i+1} \\cup \\bigcup_{j \\in [t_i]} V(C_j^i)$. Let us verify that the conditions of Lemma \\ref{lemma:absorbing} are met. First, $|W_i| = n\/2 - |U_{i+1}|$ and\n  $$\n    |U_{i+1}| = \\sum_{j = i+1}^{4k-1} t_j \\cdot (r \\ell_j + 1) < n\/4.\n  $$\n  From \\eqref{eq:ell_i_t_i} we have $(\\ell_i + 1) t_{i+1} > n^{1 - \\beta\/2}$,  and as\n  $$\n    (t_i + t_{i+1})(r \\ell_i + 1) < 2 t_i (r \\ell_i + 1) < n \/ 16 < |W|\/2\n  $$\n  we can indeed apply Lemma \\ref{lemma:absorbing} in each iteration. \n\n  Finally, let $W_0 = V_2 \\setminus U_1$. Apply Lemma \\ref{lemma:absorbing} one last time with $\\ell_0 = 0$, $t_0 = |W_0|\/4$ and $C_1^1, \\ldots, C_{t_1}^1$ (as $C_1', \\ldots, C_{t'}')$. This is justified as $t_1=\\Theta(n^{1-1\/4k})$ and $t_0+t_1=o(n)$. The obtained $0$-chains are then just a set of vertices $C_0 \\subseteq W_0$ with the property that for every $L_0 \\subseteq C_0$ there exists a subset $L_1' \\subseteq [t_1]$ such that the subgraph of $G$ induced by\n  $$\n    L_0 \\cup \\left( \\bigcup_{j \\in L_1'} V(C_j^1) \\right)\n  $$  \n  contains a $K_r$-factor. \n\n  Next, we show that the set $C_0 \\cup U_1$ has a strong absorbing property.\n  \\begin{claim} For any subset $L_0 \\subseteq C_0$ such that $|L_0| + |U_1| \\in r \\mathbb{Z}$, the induced subgraph $G[L_0 \\cup U_1]$ contains a $K_r$-factor.\n  \\end{claim}\n  \\begin{proof}\n  Consider one such $L_0$ and let $L_1' \\subseteq [t_1]$ be a subset such that\n  $$\n    L_0 \\cup \\left(\\bigcup_{j \\in L_1'} V(C_j^1) \\right)\n  $$\n  contains a $K_r$-factor. We further take $L_1 = [t_1] \\setminus L_1'$ and use the property guaranteed by Lemma \\ref{lemma:absorbing} to obtain a subset $L_2' \\subseteq [t_2]$ such that the subgraph of $G$ induced by \n  $$\n    \\left( \\bigcup_{j \\in L_1} V(C_j^1)  \\right) \\cup \\left( \\bigcup_{j \\in L_2'} V(C_j^2) \\right)\n  $$\n  contains a $K_r$-factor. Continuing this way, we obtain a subset $L_{4k-1}' \\subseteq [t_{4k - 1}]$ such that the subgraph of $G$ induced by \n  $$\n    L_0 \\cup \\bigcup_{i = 1}^{4k - 2} \\left( \\bigcup_{j \\in [t_i]} V(C_j^i) \\right)  \\cup \\left( \\bigcup_{j \\in L_{4k-1}'} V(C_j^{4k - 1}) \\right) = (L_0 \\cup U_1) \\setminus \\bigcup_{j \\in L_{4k - 1}} V(C_j^{4k - 1})\n  $$\n  contains a $K_r$-factor, where $L_{4k - 1} = [t_{4k - 1}] \\setminus L_{4k - 1}'$. As $|V(C_j^{4k - 1})| \\equiv 1 (\\textrm{mod } r)$ and $|L_0| + |U_1| \\in r \\mathbb{Z}$ we necessarily have $|L_{4k - 1}| \\in r \\mathbb{Z}$. Therefore, to complete a $K_r$-factor in $G[L_0 \\cup U_1]$ it suffices to partition $L_{4k - 1}$ into groups of size $r$ and for each such group $\\{i_1, \\ldots, i_r\\}$ find a copy of $K_r$ with one vertex in each $R(C_{i_1}^{4k - 1}), \\ldots, R(C_{i_r}^{4k - 1})$ (see Observation \\ref{obs:canonical}). The existence of such $K_r$ follows from \\ref{prop:chain} and $|R_j^{4k - 1}| = \\ell_{4k - 1} + 1 > n^{1 - 1\/4k} > n^{1 - \\beta}$. \n  \\end{proof}\n\n  We now use this absorbing property to find a $K_r$-factor in $G$. First, let $B \\subseteq V_1 \\cup (W_0 \\setminus C_0)$ be the set of all vertices which are not part of chains and such that they have less than $|C_0|p\/2$ neighbours in $C_0$. As $|C_0| \\ge \\alpha n$, we have $|B| < \\log n \/ p \\ll np$, by~\\ref{prop:expand} and the lower bound on $p.$ By \\ref{prop:in_nbr} and the assumption that every vertex has at least $np\/8$ neighbours in $V_1$, we can iteratively take one vertex $v \\in B$ at a time and find a copy of $K_r$ which contains $v$ and has all other vertices in $V_1 \\setminus B$. This takes care of $B$. Furthermore, we can continue covering the remaining vertices in $V_1 \\cup (W_0 \\setminus C_0)$ (i.e.\\ those which are not part of previously chosen $K_r$'s) with disjoint copies of $K_r$ as long as there are still at least $r n^{1 - \\beta}$ vertices, by~\\ref{prop:chain}. Let us denote the set of remaining vertices by $L$. With the absorbing property of $C_0 \\cup U_1$ in mind, to find a $K_r$-factor of $G$ it now suffices to find vertex-disjoint copies of $K_r$, each of which contains one vertex from $L$ and the others from $C_0$. Whatever we are left with in $C_0$ is guaranteed to form a $K_r$-factor with $U_1$, thus we are done. Note that this is very similar with how we took care of $B$, however the main difference is that $L$ is significantly larger than $B$ and a simple greedy strategy might not work. Instead, we find the desired copies of $K_r$ using Haxell's matching theorem (Theorem~\\ref{thm:haxell}). \n\n  For each $v \\in L$ create an $(r-1)$-uniform hypergraph $H_v$ on the vertex set $C_0$ such that $\\{v_1, \\ldots, v_{r-1}\\}$ forms a hyperedge if and only if $\\{v, v_1, \\ldots, v_{r-1}\\}$ form $K_r$ in $G$. If we can find for each $v \\in L$ a hyperedge $h_v \\in E(H_v)$ such that all these hyperedges are pairwise vertex-disjoint, then we are done.\n  To show that such edges exist it suffices to verify Haxell's criterium:\n  \\begin{equation} \\label{eq:verify_haxell}\n    \\tau(\\bigcup_{v \\in I} H_v) \\ge 2(r-1)|I|\n  \\end{equation}\n  for every $I \\subseteq L$. \n  Equivalently, for all subsets $I\\se L$ and all $Z \\subseteq C_0$ of size $|Z| \\le 2(r-1)|I|$  there exists a copy of $K_r$ with one vertex in $I$ and all other vertices in $C_0 \\setminus Z$.\n\n  We consider two cases. Consider first the case when $|I| \\le \\log n \/p$ and let $Z$ be some subset of $C_0$ of size at least $2r \\log n \/ p$. As $L \\cap B = \\emptyset$, every vertex $v \\in I$ has at least $|C_0|p\/2 > np \/ 32$ neighbours in $V_1$, thus the subset $X = (N_G(v) \\cap C_0) \\setminus Z$ is of size at least $np\/16$ (we used $np \\gg \\log n \/p$ which follows from the lower bound on $p$). By \\ref{prop:in_nbr} there exists a copy of $K_{r-1}$ in $X$. Suppose now that $|L| \\ge |I| > \\log n \/p$ and consider a subset $Z \\subseteq C_0$ of size $2r |L| < n\/32$. The set $Y = C_0 \\setminus Z$ is then of size at least $n\/16$. Thus, there exists a vertex $v \\in I$ with at least $|Y|p\/2 \\ge np\/32$ neighbours in $Y$,  by \\ref{prop:expand}. By \\ref{prop:in_nbr} such a neighbourhood contains a copy of $K_{r-1}$, which gives us a desired copy of $K_r$. This finishes the proof.\n\\end{proof}\n\nIt remains to prove Lemma \\ref{lemma:absorbing}.\n\n\\begin{proof}[Proof of Lemma \\ref{lemma:absorbing}]\n  By repeated application of Lemma \\ref{lemma:long_chain} we can find a collection of $t + t'$ disjoint $(r, \\ell)$-chains $C_1, \\ldots, C_{t + t'} \\in G[W]$. Clearly, for this we could have allowed $W$ to be much smaller than $n\/8$, thus this constraint is only for convenience. For each $i \\in [t + t']$ we create an auxiliary $(r-1)$-uniform hypergraph $H_i$ on the vertex set $V' = [t']$ by adding a hyperedge $\\{j_1, \\ldots, j_{r-1}\\}$ if and only if there exists a copy of $K_r$ in $G$ with one vertex in each $R(C_i), R(C_{j_1}'), \\ldots, R(C_{j_{r-1}}')$. Note that for every such hyperedge the subgraph of $G$ induced by \n  $$\n    V(C_i) \\cup V(C_{j_1}') \\cup \\ldots V(C_{j_{r-1}}')\n  $$\n  contains a $K_r$-factor (see Observation \\ref{obs:canonical}).\n\n  We first show that there exists a subset $B \\subseteq [t + t']$ of size at most $|B| \\le t'$ such that for every subset $J \\subseteq [t + t'] \\setminus B$ of size $|J| \\le t'\/8r$ we have\n  \\begin{equation} \\label{eq:2rJ}\n    \\tau(\\bigcup_{i \\in J} H_i) \\ge 2r|J|.\n  \\end{equation}\n  Initially, set $q = 0$ and $B = \\emptyset$. As long as $|B| < t'\/8r$ and there exists a subset $J \\subseteq [t + t'] \\setminus B$ of size $|J| \\le t'\/8r$ that violates \\eqref{eq:2rJ} set $B = B \\cup J$, $J_{q+1} = J$ and increase $q$ by 1. Suppose towards a contradiction that for some $q$, $|B| \\ge t'\/8r$, and let $q$ be the smallest such index. Then $|B| \\le t'\/4r$ as $|J_q| \\le t'\/8r$. Moreover, we have\n  $$\n    \\tau(\\bigcup_{i \\in B} H_i) \\le \\sum_{j = 1}^q \\tau(\\bigcup_{i \\in J_j} H_i) < \\sum_{j = 1}^q 2 r |J_j| = 2r|B| \\le t'\/2.\n  $$\n  This implies that there exists a set $\\tilde B \\se V'$ of size at most $t'\/2$ such that every hyperedge $h\\in\\bigcup_{i\\in B}H_i$ intersects $\\tilde B$. In other words, there exists $B' \\subseteq V'$ of size $|B'| \\ge t'\/2$ such that there is no copy of $K_r$ in $G$ with one vertex in $\\bigcup_{i \\in B} R(C_i)$ and the others in each $R(C_{j_2}'), \\ldots R(C_{j_{r}}')$ for some distinct $j_2, \\ldots, j_{r} \\in B'$. Split $B'$ arbitrarily into $r-1$ sets of nearly equal size, denoted by $B_2', \\ldots, B_{r}'$, each of size at least $t'\/2r$, and set $X_j = \\bigcup_{i \\in B'_j} R(C_i')$ for $j=2,\\ldots,r$. Then each such $X_j$ is of size  at least\n  $$\n  (\\ell' + 1) \\frac{t'}{2r} > (\\ell + 1) \\frac{t'}{2r}.\n  $$\n  On the other hand, $X_1$ defined as $\\bigcup_{i \\in B} R(C_i)$ is of size at least \n  $$\n    |X_0| \\ge (\\ell + 1) |B| \\ge (\\ell + 1) \\frac{t'}{8r}.\n  $$\n  Thus we have $|X_i| \\ge n^{1 - \\beta}$ by the assumption of the lemma, with room to spare. By \\ref{prop:chain} there exists a copy of $K_r$ intersecting each $X_i$, which is a contradiction. Therefore we have that there exists a set  $|B|$ of size less than $t'\/8r$ and every subset $J \\subseteq [t + t'] \\setminus B$ of size $|J| \\le t' \/ 8r$ satisfies \\eqref{eq:2rJ}.\n\n  Take an arbitrary $t$-subset $I \\subseteq [t + t'] \\setminus B$ and relabel $\\{C_i\\}_{i \\in I}$ as $\\{C_i\\}_{i \\in [t]}$. We show that such $(r, \\ell)$-chains have the desired property. Consider some $L \\subseteq [t]$. First, let $S \\subseteq L$ be a smallest subset such that the subgraph of $G$ induced by\n  $$\n    \\bigcup_{i \\in L \\setminus S} V(C_i)\n  $$\n  contains a $K_r$-factor. We claim that $|S| < t'\/8r$. Suppose towards a contradiction that $|S| \\ge t' \/ 8r$. Consider an equipartition $S = S_1 \\cup \\ldots \\cup S_r$. Then each set $X_i = \\bigcup_{j \\in S_i} R(C_j)$ is of size \n  $$\n    |X_i| \\ge (\\ell + 1) \\frac{t'}{8r^2} > n^{1 - \\beta}\n  $$\n  thus by \\ref{prop:chain} there exist a copy of $K_r$ intersecting each $X_i$. Therefore there exists distinct $i_1, \\ldots, i_r \\in S$ and a copy of $K_r$ intersecting each $R(C_{i_j})$. By Observation \\ref{obs:canonical} this is a contradiction with the minimality of $S$. Finally, as $|S| \\le t'\/8r$ we have that every subset $J \\subseteq S$ satisfies \\eqref{eq:2rJ} thus we can choose $h_i \\in H_i$ for each $i \\in S$ such that these edges are pairwise vertex disjoint, by Theorem~\\ref{thm:haxell}. Let $L' = \\bigcup_{i \\in S} h_i$. The construction of such hyperedges implies that the subgraph of $G$ induced \n  $$\n    \\bigcup_{i \\in L} V(C_i) \\cup \\bigcup_{i \\in L'} V(C_i')\n  $$\n  contains a $K_r$-factor, as desired.\n\\end{proof}\n\n\n\n\\section{Breaker's strategy}\n\nThe idea behind the proof is that Breaker prevents a fixed vertex $v$ from being in a copy of $K_r$ in Maker's graph. To illustrate why this could be possible, fix a vertex $v\\in [n]$ and assume for now that Maker at first only claims edges incident to $v$, as long as there is at least one such unclaimed edge. Breaker responds by claiming $b$ edges incident to $v$ in every round as well, so that at the end of this first stage of the game the set of neighbours of $v$ in Makers graph, denoted by $N_M(v)$, has size roughly $n\/b$. For the rest of the game, Breaker only needs to prevent Maker from claiming a copy of $K_{r-1}$ in $N_M(v)$, which is possible if $b\\ge C (n\/b)^{2\/r}$ for some constant $C$ which is independent of $n$, by the result of Bednarska and \\L uczak~\\cite{bl2000}; or equivalently if $b\\ge C n^{2\/(r+2)}$ (with a different constant $C$). \n\nIf Maker indeed first claims as many edges incident to $v$ as possible, this would be the end of the proof. Of course, we cannot rely on this assumption. The way to counterfeit it is to divide the attention of Breaker into two: the first $b\/2$ claimed edges are incident to $v$, thus preventing its neighbourhood in the Maker's graph from becoming larger than $2n\/b$; the second $b\/2$ claimed edges lie inside its current neighbourhood and prevent a copy of $K_{r-1}$. Crucially, the board of the game where we want to use the strategy ${\\mathcal S}$ from~\\cite{bl2000} will be revealed over time only (as the neighbourhood of $v$ in Maker's graph increases). It turns out that the proof of a static version of the game (where the whole board is `visible') can be turned into a proof of a suitable dynamic version (where the board is revealed over time). Unfortunately, none of the ingredients of the proof is black-boxable so we need to dig into each part.  \n\nLet us introduce necessary notation in a bit more generality than needed for our application. Let ${\\mathcal H}$ be a given hypergraph, say on vertex set $V({\\mathcal H})$ and edge set $E({\\mathcal H})$, and let $m$ and $b$ be integers. We define the {\\em dynamic-board $({\\mathcal H},m,b)$-game} as follows. \nLet $V_0 = \\emptyset$. The two players Maker and Breaker play in rounds, with Maker going first. For $i\\ge 0$, suppose that $i$ rounds have been played and that $V_i\\se V({\\mathcal H})$ is defined. In round $i+1$, Maker may play either according to {\\em Option (a)} in which she claims up to $m$ elements of $V_i$ and sets $V_{i+1}=V_i$ (in case there are less than $m$ elements she claims all of them), or according to {\\em Option (b)} in which she chooses elements $v_1,\\ldots,v_{\\ell}$ (for some $\\ell\\ge 1$) from $V({\\mathcal H}) \\sm V_i$ and sets $V_{i+1} = V_i\\cup\\{v_1,\\ldots,v_{\\ell}\\}$ (but Maker does not claim edges in a round when she enlarges the board). In case Option (a) is not possible, Maker is forced to play Option (b), unless it is the end of the game. Afterwards, Breaker claims (up to) $b$ elements in $V_{i+1}$. In case there are less than $b$ unclaimed elements in $V_{i+1}$, Breaker claims all of them. Maker wins if at the end of the game she has claimed all elements of some hyperedge $H\\in E({\\mathcal H})$. Otherwise, Breaker wins. \n\nGiven a (fixed) graph $H$ and a complete graph $K_n$, we define a {\\em dynamic $b$-biased $H$-game} as the $({\\mathcal H}, 1, b)$-game where the vertex set of ${\\mathcal H}$ are the edges of $K_n$, and the hyperedges of ${\\mathcal H}$ correspond to edge sets of $K_n$ which form a copy of $H$. The following theorem is a generalisation of the mentioned result by Bednarska and \\L uczak \\cite{bl2000} to the dynamic setting. For the definition of $m_2(H)$, see Theorem \\ref{thm:KLR}.\n\n\\begin{theorem}\\thlab{ourDynamicBreaker}\nFor every graph $H$ which contains at least three non-isolated vertices there exists a constant $C>0$ such that Breaker has a winning strategy in the dynamic $b$-biased $H$-game played on $K_n$ if $b\\ge Cn^{1\/m_2(H)}$.\n\\end{theorem}\n\n\nWe may take $C$ sufficiently large such that the theorem statement is true for small $n$, so that in the proof we can safely assume that $n$ is as large as needed. The proof of Theorem~\\ref{ourDynamicBreaker} proceeds along the lines of~\\cite{bl2000}. We sketch the argument in the next section, leaving out calculations that are identical to those in~\\cite{bl2000}. \n\nTheorem \\ref{ourDynamicBreaker} is all we need to describe Breaker's strategy for isolating a vertex $v$ from being in a copy of $K_r$. \n\n\\begin{proof}[Proof of Theorem \\ref{thm:main} (ii)]\nLet $r\\ge 4$, let $C$ be a large enough constant, let $n$ be an integer and let $b\\ge Cn^{2\/(r+2)}$. Let $v$ be a fixed vertex of $K_n$. We show that Breaker has a strategy in the $b$-biased Maker--Breaker game played on the edge set of $K_n$ to prevent Maker from claiming a copy of $K_r$ that contains the vertex $v$. Consequently, Maker's graph does not contain a $K_r$-factor. Before we present the strategy of Breaker we describe an auxiliary game that Breaker simulates in parallel. \n\nLet $T$ be a set of size $2n\/b$, disjoint from $V(K_n)$. By \\thref{ourDynamicBreaker}, if $b\/2\\ge C'(2n\/b)^{1\/m(K_{r-1})}$ then Breaker has a winning strategy ${\\mathcal S}$ in the dynamic $(b\/2)$-biased $K_{r-1}$-game played on $K_T$, the complete graph on the vertex set $T$. Equivalently, $b\\ge Cn^{2\/(r+2)}$ for suitable $C$. \n\nWe now describe the strategy of Breaker in the $b$-biased Maker--Breaker game played on the edge set of $K_n$. Suppose that $i \\ge 0$ rounds have been played already. Let $M$ and $B$ denote the graphs formed by Maker's edges and by Breaker's edges, respectively (we suppress dependence on $i$ for clarity of presentation). Breaker maintains the property that every vertex $w\\in N_M(v)$ has a (unique) corresponding vertex $t_w\\in T$ such that an edge $uw$ in $N_M(v)$ belongs to Maker's (Breaker's) graph if and only if $t_ut_w$ belongs to Maker's (Breaker's) graph in the auxiliary dynamic $b$-biased $K_{r-1}$-game played on $K_T$. Clearly, this is the case before the first round of the game, and we show that Breaker can maintain such a correspondence throughout the game. Set $T_{i} = \\{t_w:w\\in N_M(v)\\}$.  \n\nLet $xy$ denote the edge that Maker claims in round $i+1$. Then Breaker claims up to $b\/2$ edges incident to $v$ including $xv$ or $yv$ if those edges are not claimed yet by either of the players. If Breaker has claimed $b' < b\/2$ edges and there are no more unclaimed edges incident to $v$, then he claims $b\/2 - b'$ arbitrary edges (note that additional edges do not hurt Breaker). For the remaining $b\/2$ edges in round $i+1$ we distinguish between three cases (where the latter two are similar). In Case~1, assume that $x\\not\\in N_M(v) \\cup \\{v\\}$ or $y\\not\\in N_M(v) \\cup \\{v\\}$. Then Breaker claims $b\/2$ arbitrary edges. In Case~2.1, assume that $x=v$ (the case $y=v$ is analogous). Let $t\\in T\\sm T_{i}$ and set $t_y= t$. In the auxiliary dynamic $K_{r-1}$-game, Breaker pretends that (the auxiliary) Maker plays according to Option (b) and adds the elements $\\{t_y  t_w: w\\in N_M(v)\\}$ to the board (recall that the vertices in the hypergraph corresponding to that game are the edges of $K_T$). In Case~2.2, assume that $x,y\\in N_M(v)$. Then Breaker pretends that in the auxiliary dynamic $K_{r-1}$-game Maker plays according to Option (a) and claims the edge $t_xt_y$. In either of Case~2.1 or~2.2, the strategy ${\\mathcal S}$ in the auxiliary game gives $b\/2$ edges $e_1,\\ldots, e_{b\/2} \\in E(K_T)$ for Breaker to claim in the auxiliary board. Let $f_1,\\ldots,f_{b\/2}$ be the corresponding edges in $N_M(v)$, that is $f_i$ is the edge with endpoints $w_i$ and $u_i$ such that $e_i$ has endpoints $t_{w_i}$ and $t_{u_i}$. Breaker then claims $e_1,\\ldots, e_{b\/2}$ in the auxiliary game and $f_1,\\ldots,f_{b\/2}$ in the real game. \n\nWe claim that this is indeed a winning strategy. First note that $|N_M(v)|\\le 2n\/b$ since Breaker claims $b\/2$ of the $n-1$ total edges incident to $v$ in every round. Thus, the set $T$ is large enough so that Breaker can indeed maintain an injective map $w\\mapsto t_w$ for $w\\in N_M(v)$. Furthermore, it is clear from the strategy description that a Maker\/Breaker edge in $N_M(v)$ corresponds to a Maker\/Breaker edge in the auxiliary game in $T$. Finally, since ${\\mathcal S}$ is a strategy for Breaker to prevent Maker in the auxiliary $b\/2$-biased game to claim a copy of $K_{r-1}$ this implies that Breaker can indeed prevent Maker from claiming a copy of $K_{r-1}$ in $N_M(v)$, i.e.~the vertex $v$ is not in a copy of $K_r$ in Maker's graph.\n\\end{proof}\n\n\n\n\n\\subsection{Proof of Theorem \\ref{ourDynamicBreaker} (sketch)}\n\\label{proof:ourLemma5}\n\nThe following is a dynamic-board variant of \\cite[Lemma 5]{bl2000}. We switch notation from $m$ to $p$ and from $b$ to $q$ for the bias of Maker and Breaker, respectively, to be consistent with the literature. \n\n\\begin{lemma}\\thlab{ourLemma5} In every dynamic-board $({\\mathcal H},p,q)$-game Breaker has a strategy such that at the end of the game at most $(1+q) f({\\mathcal H},p,q)$ edges of the hypergraph ${\\mathcal H}$ have all their vertices claimed by Maker, where $f({\\mathcal H},p,q)=\\sum_{H\\in E({\\mathcal H})}(1+q)^{-|H|\/p}.$ \n\\end{lemma}\n\nThe proof is a simple adaptation of the potential function technique as introduced by Erd\\H{o}s and Selfridge~\\cite{es1973} that was generalised by Beck~\\cite{b1982} to biased Maker--Breaker games. We are unaware of such a dynamical-board variant thus the full proof follows. We follow notation and strategy of the proof of \\cite[Theorem 20.1]{BeckBook}. \n\n\\begin{proof}[Proof of \\thref{ourLemma5}]\nLet ${\\mathcal H}$, $p$, $q$ be as in the lemma and let $\\mu$ be defined by $1+\\mu = (1+q)^{1\/p}$. \nGiven two disjoint subsets $M$ and $B$ of the board $V=V({\\mathcal H})$ and an element $z\\in V$ set  \n\\begin{align*}\n\\Phi(M,B) &= \\sum_{H\\in {\\mathcal H}: H\\cap B =\\emptyset} (1+\\mu)^{-|H\\sm M|},\\text{ and}\\\\\n\\Phi(M,B,z) &= \\sum_{z\\in H\\in {\\mathcal H}: H\\cap B =\\emptyset} (1+\\mu)^{-|H\\sm M|}\n\\end{align*}\nand note straight away the following inequalities:\n\\begin{align}\n\\Phi(M\\cup\\{e\\},B,z)&\\le (1+\\mu)\\Phi(M,B,z),\\lab{T1}\\\\\n\\Phi(M,B\\cup\\{e\\},z)&\\le \\Phi(M,B,z).\\lab{T2}.\n\\end{align}\nFor integers $r$ and $j$, let \n$b_r^{(j)}$ be the $j^{\\mathrm{th}}$ element that Breaker picks in round $r$, and let $m_r^{(j)}$ be the $j^{\\mathrm{th}}$ element that Maker picks in round $r$ if she decides to play according to Option (a) and pick elements in $V_{r-1}$ rather than enlarging $V_r$ (Option (b)). \nFurthermore, let $M_r$ and $B_r$ be the set of all elements of Maker and of Breaker, respectively, {\\em after} round $r$, \nand let \n$M_{r,j}=M_r\\cup \\{m_{r+1}^{(1)},\\ldots,m_{r+1}^{(j)}\\}$ and \n$B_{r,j}=B_r\\cup \\{b_{r+1}^{(1)},\\ldots,b_{r+1}^{(j)}\\}$ (assuming that Maker\/Breaker has claimed at least $j$ elements in round $r+1$). \n\nWe now describe Breaker's strategy in round $r$. If there are less than $q$ unclaimed elements in $V_{r+1}$, then Breaker claims all of them. Otherwise, for every $1\\leq j\\le q$, sequentially, Breaker calculates $\\Phi(M_r,B_{r-1,j-1},z)$ for every unclaimed element $z\\in V_r\\sm(M_r\\cup B_{r-1,j-1})$ and claims the element $b_r^{(j)}$ which maximises this expression. Note that here we chose the element $b_r^{(j)}$ in $V_r$, and not in the whole board $V$. If, for some $j$, there are no unclaimed elements, then it is the end of Breaker's turn.\n\nThe crucial part of the potential function technique in positional games is to show that the {\\em potential} $\\Phi(M_{r+1},B_r)$ is decreasing (if evaluated after Makers move). But this is now straight-forward along the lines of the proof in \\cite{BeckBook}. The only thing we have to notice is that in round $r+1$, if Maker chooses to claim elements in $V_r$, then their choices are on the same sub-board where Breaker claimed their elements in round $r$. \n\n\\begin{claim}\nFor all $r\\ge 1$, \n$\\Phi(M_{r+1},B_r)\\le \\Phi(M_r,B_{r-1})$.\n\\end{claim}\n\n\\begin{proof} \nIf Maker has played according to Option (b) in round $r+1$, then $\\Phi(M_{r+1}, B_r) = \\Phi(M_r, B_r) \\le \\Phi(M_r, B_{r-1})$, where the inequality follows from \\eqref{T2}. Therefore, if Breaker was not able to claim $q$ elements in round $r$, then in round $r+1$ Maker is forced to play Option (b) and the claim follows. For the rest of the proof we can assume that Breaker is able to claim $q$ elements in round $r$. Without loss of generality, we can also assume that Maker claims $p$ elements in round $r+1$ (claiming fewer than $p$ elements only makes it easier for the desired inequality to hold). Let us denote these elements by $b_r^{(1)}, \\ldots, b_r^{(q)}$ and $m_{r+1}^{(1)}, \\ldots, m_{r+1}^{(p)}$, respectively.\n\nWe first note that $\\Phi(M_r,B_{r-1,j+1})=\\Phi(M_r,B_{r-1,j})-\\Phi(M_r,B_{r-1,j},b_r^{(j+1)})$ for all $0\\le j<q$, and \n $\\Phi(M_{r,j+1},B_{r})=\\Phi(M_{r,j},B_{r})+\\mu\\Phi(M_{r,j},B_{r},m_{r+1}^{(j+1)})$ for all $0\\le j<p$.  Hence, \n\\begin{align*}\n\\Phi(M_{r+1},B_r)&= \\Phi(M_r,B_{r-1}) \n\t- \\sum_{j=1}^q \\Phi(M_r,B_{r-1,j-1},b_r^{(j)}) \n\t+\\mu \\sum_{j=1}^p \\Phi(M_{r,j-1},B_{r},m_{r+1}^{(j)}).\n\\end{align*}\nIt suffices to show  \n\\begin{align}\n\\lab{aux343}\n&\\mu \\sum_{j=1}^p \\Phi(M_{r,j-1},B_{r},m_{r+1}^{(j)}) \\le \\sum_{j=1}^q \\Phi(M_r,B_{r-1,j-1},b_r^{(j)}).\n\\end{align}\nFirst note that for all $1\\le j\\le p$, \n$$\\Phi(M_{r,j-1},B_{r},m_{r+1}^{(j)})\\le (1+\\mu)^{j-1}\\Phi(M_{r},B_{r},m_{r+1}^{(j)})\\le (1+\\mu)^{j-1}\\Phi(M_{r},B_{r-1,q-1},m_{r+1}^{(j)}),$$\nwhere the first inequality follows from~\\eqref{T1} and the second from~\\eqref{T2}. \nFurthermore, the element $b_r^{(q)}$ is chosen by Breaker {\\em before} Maker claims any of $m_{r+1}^{(j)}$  \nand it is chosen to maximise the expression $\\Phi(M_{r},B_{r-1,q-1},z)$ \n{\\em over all unclaimed $z\\in V_{r+1}$} (note that Maker sets $V_{r+1}$ after her move in round $r$). Hence, we deduce \n\\begin{align*\n\\Phi(M_{r,j-1},B_{r},m_{r+1}^{(j)})\\le (1+\\mu)^{j-1}\\Phi(M_{r},B_{r-1,q-1},b_r^{(q)}), \n\\end{align*}\nwhich readily implies that \n\\begin{align}\n\\lab{aux346}\n\\mu \\sum_{j=1}^p \\Phi(M_{r,j-1},B_{r},m_{r+1}^{(j)}) \\le \\mu \\Phi(M_{r},B_{r-1,q-1},b_r^{(q)}) \\sum_{j=1}^p (1+\\mu)^{j-1}.\n\\end{align}\nTo bound the right hand side of~\\eqref{aux343} we note that for all $1\\le j < q$, \n\\begin{align*\n&\\Phi(M_{r},B_{r-1,j-1},b_r^{(j)})\\ge \\Phi(M_{r},B_{r-1,j-1},b_r^{(q)})\\ge \\Phi(M_{r},B_{r-1,q-1},b_r^{(q)}),\n\\end{align*}\nwhere the first inequality follows again by $b_r^{(j)}$ maximising $\\Phi(M_{r},B_{r-1,j-1},z)$ among all unclaimed elements $z$ in $V_{r+1}$ (and $b_r^{(j)}$ is chosen before $b_r^{(q)}$), and the second inequality follows from~\\eqref{T2}. \nThis implies that \n$$\\sum_{j=1}^q \\Phi(M_{r},B_{r-1,j-1},b_r^{(j)})\\ge q  \\Phi(M_r,B_{r-1,q-1},b_r^{(q)}).$$\nThis and~\\eqref{aux346} imply that the inequality in~\\eqref{aux343} follows from \n$$\\mu \\Phi(M_{r},B_{r-1,q-1},b_r^{(q)}) \\sum_{j=1}^p (1+\\mu)^{j-1}\n\\le q \\Phi(M_r,B_{r-1,q-1},b_r^{(q)}),$$ \nwhich is true since $\\Phi(M_{r},B_{r-1,q-1},b_r^{(q)})\\ge 0$ and $(1 + \\mu)^p - 1 = q$.\n\\end{proof} \n \nTo finish the proof of  \\thref{ourLemma5} note that \n$\\Phi (\\emptyset,\\emptyset) = f({\\mathcal H},p,q)$ and that after Maker's move in the first round we have \n$\\Phi(M_1,B_0) = \\Phi(M_1,\\emptyset) \\le (1+\\mu)^p f({\\mathcal H},p,q) = (1 + q) f({\\mathcal H}, p ,q)$. \nBy the above claim, the function $f(M,B)$ is decreasing if evaluated after Maker's move. Hence, at the end of the game, we have \n$\\Phi(M,B)\\le \\Phi(M_1,B_0)\\le (1+q) f({\\mathcal H},p,q)$, where now $M$ and $B$ denote the final set of elements that Maker and Breaker claimed, respectively. On the other hand, if Maker occupies all elements of an edge $H\\in E({\\mathcal H})$ (at any point of the game), then the additive contribution to $\\Phi(M,B)$ is 1 for each such set. \nThus, if $(1+q)f({\\mathcal H},p,q)<k$ then Breaker has a strategy in the dynamic Maker--Breaker game so that at the end of the game Maker fully occupies fewer than $k$ sets $H\\in E({\\mathcal H})$. \n\\end{proof}\n\nA complete proof of \\thref{ourDynamicBreaker} including all details would involve repeating the proof of \\cite[Theorem~1~(ii)]{bl2000} and replacing \\cite[Lemma 5]{bl2000} by \\thref{ourLemma5} wherever it occurs. Instead, we give an overview of the argument in~\\cite{bl2000} and describe where the changes are needed. We leave out calculations that are identical. \n\nWithout loss of generality we may assume that $H$ is such that $m_2(H)$ achieves maximum for $H' = H$, i.e.\\ $m_2(H) = (e(H) - 1)\/(v(H) - 2)$. If this is not the case, then Breaker can choose $H' \\subseteq H$ which determines $m_2(H)$ (which might not be unique), and prevent Maker from creating $H'$. Call such an $H$ {\\em $m_2$-maximal.} \n\nBednarska and \\L uczak first identify the dangerous structures that Maker can create during a $b$-biased $H$-game on $K_n$. We say that a graph $F$ is an $\\bar H$-graph if it contains two vertices $v$, $w$ such that $F+vw$ is isomorphic to $H$. We shall write $F^{vw}$ to specify such vertices in $F$. Let ${\\mathcal F}=\\{F_1^{v_1w_1},\\ldots,F_t^{v_tw_t}\\}$ be a family of different $\\bar H$-graphs, whose vertex sets may intersect. Then ${\\mathcal F}$ is called a {\\em $t$-fan} if $|\\bigcap_{i=1}^t V(F_i^{v_iw_i})|\\ge 2$, and it is called a {\\em $t$-flower} if $|\\bigcap_{i=1}^t V(F_i^{v_iw_i})|\\ge 3$. Furthermore, a $t$-fan ${\\mathcal F}$ is called {\\em simple} if $|\\bigcap_{i=1}^t V(F_i^{v_iw_i})|=2$. If at some point during the game Maker's graph contains some $\\bar H$-graph $F^{vw}$ such that the pair $\\{v,w\\}$ has not been claimed by Breaker, then $F^{vw}$ is {\\em dangerous}. Similarly, a $t$-fan ${\\mathcal F}$ is called {\\em dangerous} if its elements are dangerous subgraphs of Maker's graph. \n\nThe following is the dynamic-board variant of~\\cite[Lemma 9]{bl2000}.\n\\begin{lemma}\\thlab{ourLemma9}\nFor every $m_2$-maximal graph $H$ that contains a cycle there exist positive constants $C$ and $n_0$ such that for every $n\\ge n_0$ and $q\\ge Cn^{1\/m_2(H)}$ Breaker has a strategy such that at no stage of the dynamic $q$-biased $H$-game on $K_n$ Maker's graph contains a dangerous $q$-fan. \n\\end{lemma}\nBefore sketching the proof of this lemma let us briefly explain how it implies \\thref{ourDynamicBreaker}. \nWhen $H$ is a forest the argument is identical to the one given in~\\cite{bl2000}: The requirement $b\\ge 2n^{1\/m_2(H)}$ implies that Breaker has a simple strategy in either case of~$H$ containing a path on three vertices or consisting of disjoint edges. \nWhen $H$ contains a cycle let $b\\ge 2Cn^{1\/m_2(H)}$ where $C$ is the constant given by \\thref{ourLemma9}. Breaker's strategy in the dynamic $b$-based $H$-game played on $K_n$ is as follows. In every round, Breaker claims $b\/2$ edges to follow the strategy of \\thref{ourLemma9} and thus prevents Maker from building a dangerous $(b\/2)$-fan. The remaining $b\/2$ choices are used to block the pairs $v_1w_1,\\ldots,v_tw_t$ of a dangerous $t$-fan $\\{F_1^{v_1w_1},\\ldots,F_t^{v_tw_t}\\}$, if there is one for some $t\\le b\/2$. Note that this is indeed a winning strategy. Suppose that Maker can claim a copy of $H$, say in round $r$ of the game. Then Maker has created a dangerous $t$-fan for some $t\\ge 1$ in round $r-1$ (or earlier). Since Maker plays with bias 1, she cannot create more than one dangerous $t$-fan per round. By \\thref{ourLemma9}, $t\\le b\/2$, thus Breaker would block such a dangerous $t$-fan in round $r-1$. \n\nWe now sketch the proof of \\thref{ourLemma9}. The strategy for Breaker is based, yet again, on two strategies that are followed each with bias $q\/2$. The following two lemmata encapsulate these strategies, they are the dynamic-board variants of Lemma~6 and Lemma~7 of~\\cite{bl2000}, respectively. \n\\begin{lemma}\\thlab{ourLemma6}\nFor every $m_2$-maximal graph $H$ that contains a cycle there exist positive constants $C_1$, $n_1$ and $\\delta <1$ such that for every $n\\ge n_1$ and $q\\ge C_1n^{1\/m_2(H)}$ Breaker has a strategy such that at each moment of the dynamic $q$-biased $H$-game on $K_n$ there are no dangerous $s$-flowers in Maker's graph, where $s=q^{1-\\delta}$. \n\\end{lemma}\n\\begin{lemma}\\thlab{ourLemma7}\nFor every $m_2$-maximal graph $H$ that contains a cycle and every positive $\\delta <1$, there exist constants $C_2$ and $n_2$ such that for every $n\\ge n_2$ and $q\\ge C_2n^{1\/m_2(H)}$ Breaker has a strategy in the dynamic $q$-biased $H$-game on $K_n$ which does not allow Maker to build $\\frac12 \\binom{q}{t}$ simple $t$-fans, where $t=q^{\\delta\/3}.$ \n\\end{lemma}\nTo prove \\thref{ourLemma9}, let $C_1$, $n_1$, $\\delta$ be given as in \\thref{ourLemma6}, let $C_2$ and $n_2$ be as in \\thref{ourLemma7}, and set $C=2\\max\\{C_1,C_2\\}$ and let $n$ be large enough. \nLet ${\\mathcal S}_1$ be the strategy given by \\thref{ourLemma6} for the dynamic $(q\/2)$-biased $H$-game on $K_n$, and let ${\\mathcal S}_2$ be the strategy given by \\thref{ourLemma7} for the dynamic $(q\/2)$-biased $H$-game on $K_n$. Then, using $q\/2$ edges to follow strategy ${\\mathcal S}_1$ and $q\/2$ edges to follow strategy ${\\mathcal S}_2$, Breaker can ensure that at no point during the dynamic $q$-biased $H$-game on $K_n$ Maker's graph contains a dangerous $s$-flower with $s= (q\/2)^{1-\\delta}$, nor $\\frac12 \\binom{q\/2}{t}$ simple $t$-fans with $t=(q\/2)^{\\delta\/3}.$ \nThe reasoning why this implies that Maker's graph never contains a dangerous $q$-fan is exactly the same as in~\\cite{bl2000}, see the last paragraph of the proof of Lemma~9 therein. \n \nFinally, we explain the proofs of \\thref{ourLemma6,ourLemma7}. \n\nFor \\thref{ourLemma6}, define a {\\em $t$-cluster} to be a graph consisting of $t$ copies of $H$ that intersect in at least 3 vertices (the difference to a $t$-flower is that here the building blocks are copies of $H$ rather than $\\bar H$-graphs). Let ${\\mathcal H}_1$ be the collection of all edge sets on $K_n$ that form a copy of a {\\em $t$-cluster}, where $t=t(H)$ is a large enough constant. \nFurther, let $q_1 = q^{1-\\delta_1\/2}$ where $\\delta_1=\\delta_1(H)$ is a suitable constant as defined at the beginning of the proof of~\\cite[Lemma 6]{bl2000}. The choice of parameters is identical to those in~\\cite[Lemma 6]{bl2000}. Hence, calculations identical to those in~\\cite[Lemma 6]{bl2000} show that $(q+1)f({\\mathcal H}_1,1,q_1)<1$. It follows from \\thref{ourLemma5} that Breaker has a strategy in the dynamic-board $({\\mathcal H}_1,1,q)$-game to prevent Maker from claiming all edges of a $t$-cluster. \nWe pause for a psychological note: This strategy does not forbid the creation of one copy of $H$ per se, which may seem contradictory to the final goal. For the strategy of \\thref{ourLemma6} this is irrelevant though and shall not concern us. \nIt then follows along the lines of the last two paragraphs of~\\cite[Lemma 6]{bl2000} that by preventing such a $t$-cluster for sufficiently large $t$ (depending only on $H$), Breaker can also prevent that, in the dynamic $q$-biased $H$-game on $K_n$, Maker claims the edge set of a dangerous $s$-flower, for $s=q^{1-\\delta}$, $\\delta = \\delta_1\/4$, and $q\\ge n^{1\/m_2(H)}.$ \n\nFor \\thref{ourLemma7}, let ${\\mathcal H}_2$ consist of the edge sets of simple $t$-fans in $K_n$, for $t$ as in the lemma statement. Following the calculations in the proof of~\\cite[Lemma 7]{bl2000} we find that $(q+1)f({\\mathcal H}_2,1,q)<\\frac12\\binom{q}{t}$ if $q\\ge C_2n^{1\/m_2(H)}$ where $C_2$ and $n$ are large enough. It follows that Breaker has a strategy in the dynamic-board $({\\mathcal H}_2,1,q)$-game such that at the end of the game at most $\\frac12\\binom{q}{t}$ of all $t$-fans are fully occupied by Maker. \n\n\n\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nGamma-ray bursts (GRB) are the most intense cosmological explosions marking the \nformation of the stellar mass compact objects (\\citealt{Meszaros_2006}). The central \nengine most probably launches a highly relativistic bipolar jet that releases most of \nthe burst energy in a few seconds to a few minutes mostly as gamma-rays, known as the\nprompt emission (\\citealt{Piran_2004_review}). The surrounding material heated by \nthe outflow glows afterwards on a much longer timescale of days to months, known as the afterglow.\nAs the prompt emission occurs close to the burst site, it carries the most important signature \nof the intrinsic properties of the outflowing plasma and the central engine. \n\nThe non thermal GRB spectral shape is primarily described by \nemission processes like, optically \nthin synchrotron emission (\\citealt{Tavani1996,Rees1994,Sari1998}), inverse\nCompton scattering (\\citealt{Ghisellini1999, Ghisellini_etal_2000}). \nThe non-thermal models alone, however, face difficulties in explaining\nmany spectral features across the GRB catalog, \nsuch as the hard low energy spectral slopes, collection of the spectral\npeaks within a small energy range of a few hundred keV and the high efficiency \nin the production of the observed gamma rays.   \nThis leads to the inclusion of thermal emission from the photosphere \n(\\citealt{Crideretal_1997, Ghirlandaetal_2003, Ryde_2004, Ryde_2005}),\nthereby representing the very standard fireball model. \nThe observed emission can be entirely from the photosphere wherein \nthe broadness is contributed by the continuous or localised subphotospheric \ndissipation of the kinetic energy or Poynting flux \n(\\citealt{Beloborodov2010, Beloborodov2013, Giannios2012, Pe'er2005, Rees&Meszaros2005, Iyyanietal_2015}),\nor may be due to the geometrical effects of the jet emission and its structure \n(\\citealt{Lundman2013,Beloborodov2011, Basak_Rao_2015_090618}).\nOr, the photospheric emission can be a sub-dominant component \n(\\citealt{Burgess_etal_2014, Axelsson_etal_2012, \nIyyani_etal_2013, Guiriecetal_2011, Guiriecetal_2013}). All the models have certain\npros and cons and confront issues in justifying the constraints obtained from observations. \nThe radiation processes of the prompt \nemission, thus, remains the most debated topic in the GRB field \n(see \\citealt{Zhang_2014_review, Kumar&Zhang2015, Pe'er2015} for  recent reviews).\n\n\nOn the other hand, the spectral analysis of the prompt emission itself is challenging because of\nthe rapid spectral evolution and multiple pulse emission which may have large overlaps.\nSeveral studies conducted for GRBs with single\/separable pulses \nshow definite evolution of the spectral peak ($E_{\\rm p}$), either hard to soft (HTS) \nor intensity tracking (IT) (\\citealt{LK_1996, Kanekoetal_2006, Luetal_2012, Basak_Rao_2014_MNRAS}). \nHowever, for GRBs with overlapping pulses, this picture is complicated which is probably affected \nby pulse overlap (e.g., \\citealt{Hakkila_Preece_2014}). \nAlso, a GRB detector having an unavoidable wide field of view, is background limited \nand suffers from poor spectral sensitivity. \n\nThe spectral study of the prompt emission has been revolutionized in the past \ntwo decades particularly after the launch of the \\emph{Swift} and the \\emph{Fermi} satellites.\nThe GRB Monitor (GBM) and the Large Area Telescope (LAT) of the \\emph{Fermi} provide seven decades of\nenergy band ranging from 8\\,keV to $\\sim200$\\,GeV, (see \\citealt{Meeganetal_2009_fermi_gbm, Atwoodetal_2009_LAT}).\nThe \\emph{Swift} Burst Alert Telescope (BAT) being a coded-mask \ninstrument provides a good localization and background estimate in 15--150\\,keV, while \nthe X-ray Telescope (XRT) covers the lower energy range  of\n0.3--10\\,keV. The localization accuracy of both together with the fast re-pointing capability \nof the spacecraft facilitates the redshift measurements for the majority of GRBs,\nsee \\cite{Gehrelsetal_2004_swift, Burrowsetal_2005_XRT}. It has been \nsuggested that the focusing instruments like the XRT, having a\nbetter sensitivity and spectral resolution than the wide-field detectors, can be used at the late \nepochs to identify any additional spectral component with a good statistical significance\n(\\citealt{Basak_Rao_2015_090618}). The novel strategy here is to exploit the specific ability\nof individual detectors at different phases of a GRB, namely, the wide band \\emph{Fermi} data at \nthe early phase, complemented by the finest spectral data of the XRT at the late prompt \nand\/or early afterglow phase (\\citealt{Basak_Rao_2015_090618}) and thereby studying the \nspectral evolution until the late epochs. For even longer GRBs, e.g., the ultra-long \nones, it is also possible to observe them with highly sensitive detectors of \\emph{Chandra} \nand wide band as well as fine spectral resolution detectors of \\emph{NuSTAR}\n(\\citealt{Basak_Rao_2015_130925A}).\n\nAnother important addition to these studies would be the polarization\nwhich can be used as a diagnostic tool between different models.\nThe first such detection was reported for GRB 021206 (linear polarization \ndegree, $\\Pi = 80 \\pm 20 \\%$), detected by \n{\\it Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI)}\n(\\citealt{Coburn&Boggs2003}, however, see \\citealt{Rutledge&Fox2004, Wigger2004}). \nThis was followed by e.g.,  {\\it INTEGRAL} observation ($100$ keV -- 1 MeV) of \nGRB 041219A and GRB 061122, \\cite{Kalemcietal_2007, McGlynnetal_2007, Gotzetal_2009, Gotzetal_2013}, \n{\\it IKAROS}\/gamma ray burst polarimeter (GAP) observation of\nGRB 100826A ($\\Pi = 25 \\pm 15 \\%$ and $31 \\pm 21 \\% $ in different time intervals), \nGRB 110721A ($\\Pi = 84^{+16}_{-28} \\%$) and GRB 110301A ($\\Pi = 70 \\pm 22 \\%$), see \n\\cite{Yonetokuetal_2011, Yonetoku_etal_2012}.  \n\nHigh polarization degree is generally owed to the observed non-thermal emission.  \nSeveral theoretical models \nhave been proposed such as Compton drag (\\citealt{Lazzati_etal_2004}), \nsynchrotron emission in ordered magnetic fields (\\citealt{Nakar2003, Granot_etal_2003}) \nas well as in random magnetic fields (\\citealt{Waxman2003}). On the other hand, \nphotospheric emission in general is thought to be unpolarised, however, \nthe photospheric emission (blackbody alone) from a structured \njet, observed off axis is predicted to show a polarization degree up to $\\Pi = 40 \\%$\n(\\citealt{Beloborodov2011, Lundmanetal_2014}).  \nIn case of subphotospheric dissipation models, polarised emission is expected\nto be observed in soft X-rays from synchrotron emission when dissipation\noccurs close to the photosphere. The MeV peak and spectrum above the\npeak is not expected to be polarised as they are formed at high optical depths\n(\\citealt{Lundman_etal_2016}).\nIn order to have a significant polarization detection, \nthe crucial factor is to have high photon statistics, \nwhich has been a concern in most of the polarization measurements. \n\n\\begin{table}\\centering\n\\caption{Observation of GRB 151006A with different detectors. Time is counted since the GBM trigger\n\\begin{tabular}{ccc}\n\n\\hline\nDetector & Time bins & Time interval (s) \\\\\n\\hline\n\\hline\nGBM      & 1--14      & $-2.0$ -- $88.1$ \\\\\nLAT-LLE  & 1--9       & $-2.0$ -- $27.5$ \\\\\nLAT ($>100$\\,MeV) & 8 and 11 & $16-20$, $37-53$ \\\\\nBAT            &  15--18        &   $88.2-253.2$               \\\\\nXRT            &  from bin 12        &   $>58.5$               \\\\\n\n\\hline\n\\end{tabular}\n\\label{tobs}\n\\end{table}\n\n \n\\begin{figure*}\\centering\n\\includegraphics[width=\\textwidth]{lc.pdf} \n\\caption{Light curve of GRB 151006A extracted from the {\\it Fermi}~, Astrosat\/CZTI and {\\it Swift}~\n at different energies (labeled in the respective panels).\n The time intervals chosen for our analysis are shown by dashed lines. \n We also mark the GBM trigger time with dot-dashed red line (at $t=0$\\,s)\n and the first LAT photon detection time with a vertical magenta line \n with horizontal dashes (at $t=17.5$\\,s). The time intervals for polarization\n measurements (0--16\\,s and 16--33\\,s) are marked in the CZTI-veto panel \n with dotted brown line.}\n\\label{lc}\n\\end{figure*}\n\n\nThe CZT Imager (CZTI), on-board the recently launched multi-wavelength Indian mission {\\it Astrosat}, \nprovides detection in the energy range 20 -- 200\\,keV and becomes an open detector above 100\\,keV, \nthereby enabling it to detect GRB events. The Veto detector when augmented, raises the detection energy \nlevel to 600 keV. Thus, when analysed along with the BAT data, the CZTI + Veto will become crucial in \nconstraining the spectral peak energies, which is otherwise not generally possible with the BAT \ndata alone (\\citealt{Raoetal_2016}). The CZTI also possesses X-ray polarization detection capabilities\nin the energy range, 100 -- 380\\,keV (\\citealt{Vadawale_etal_2015}). Thus, with its wide field \nof view, good spectral resolution and polarimetry capability, the CZTI data will be a key addition\nto the existing observatories like the {\\it Swift} and {\\it Fermi}. \n\n\n\nIn this paper, we present a detailed study of spectral evolution of \nGRB 151006A, the first detected GRB by the CZTI. We follow the strategy of multi-instrument\nanalysis using the detectors on-board \\emph{Fermi} and \\emph{Swift} at different \nphases of GRB emission. This is further complemented by \nthe polarization data from the CZTI. The paper is organized \nas follows. Section 2 presents the observation by various instruments,\nthe methodology of the data analysis and the spectral models used; section 3 presents \nthe results of the spectral fits \nand the polarization measured by CZTI, finally followed by conclusions and a discussion in Section 4.\n\n\n\n\\section{Observations and Data Analysis}\nOn 2015 October 06  at 09:54:57.83 UT, the {\\it Fermi}\/GBM (\\citealt{151006A_gcn_gbm}) triggered on GRB 151006A.\nThe burst also triggered many other detectors including the {\\it Swift}\/BAT at 09:55:01 UT (\\citealt{151006A_gcn_bat}), \nthe \\emph{Astrosat}\/CZTI (\\citealt{151006A_gcn_czti}) at 09:54:57.825 UTC, Konus-Wind at 09:54:57.7 UT \n(\\citealt{Konuswind_151006A}) and CALET at 09:54:59.97 UT (\\citealt{CALET_151006A}). \n\nIn the current paper, we present the spectral analysis of {\\it Fermi} and {\\it Swift} data, and the\npolarization measurement obtained from CZTI data.\nFigure \\ref{lc} shows a composite count rate light curve of various detectors on-board the {\\it Fermi},\n{\\it Swift}, and the CZTI of {\\it Astrosat} arranged from higher to lower energy bands. These are \nLAT (P8$\\_$SOURCE class events, $>100$ MeV), LAT Low energy \nevents (LLE, 30 MeV--100 MeV), GBM\/BGO detector (300 keV--30 MeV), \nGBM\/NaI detectors (100--300\\,keV),\nCZTI-veto (100--300\\,keV),\nGBM\/NaI detectors (8--100 keV), followed by the\n{\\it Swift} BAT in 50--100\\,keV, 25--50\\,keV, 15--25\\,keV and \nthe {\\it Swift} XRT (0.3--10\\,keV).\nThe GBM light curve shows a single pulse having a fast rise and an exponential decay (FRED) \nwith a duration ($T_{90}$) of $\\sim84$\\,s (50--300\\,keV) (\\citealt{151006A_gcn_gbm}). \nThe LLE emission of the burst is coincident with the GBM trigger and peaks simultaneously\nwith the BGO emission at $ \\sim 1.4$ s. The LLE emission is quite significant with a detection \nsigma of 54, and extends for about $\\sim 30$ s (\\citealt{151006A_LAT_gcn}). The emission consists\nof a narrow pulse accompanied by two other smaller pulses towards the end. Thus, LLE emission \ndoes not show the typical delay that is observed in its onset with respect to the GBM observations \n(\\citealt{Ackermannetal_2013_LAT}).  The LAT LLE data bridges the gap between the BGO and LAT \ndetections, and thereby helps in constraining high MeV spectral peaks \n(\\citealt{Axelsson_etal_2012,Moretti&Axelsson2016}). However, the LAT\n(P8R2$\\_$source class events, $>100$ MeV) events are observed to arrive \nonly $17.5$ s after the GBM trigger. The LAT emission is less significant with\nonly 5 photons detected in total. The top panel of Fig. \\ref{lc} shows the photons\nto be associated to the GRB with probability  $ > 0.9$ in red filled circle, \notherwise in open circles (using {\\sc gtsrcprob} of {\\it Fermi} science \ntool\\footnote{http:\/\/fermi.gsfc.nasa.gov\/ssc\/data\/analysis\/scitools\/overview.html}). \nThe delayed onset of the LAT events with respect to GBM emission is consistent\nwith that observed for long bursts as reported in \\cite{Ackermannetal_2013_LAT}.\nWe note that high energy emission in the energy range, 100 keV -- 30 MeV \nand 30 MeV -- 100 MeV, peaks nearly simultaneously at $\\sim 1.4$ s, whereas \nthe low energy emission in the energy range, 8 keV - 100 keV, peaks later at $\\sim 5.4$ s. \nThe BAT light curve also shows a single FRED like pulse with a $T_{90}$ (15 -350 keV) \nof $203.9 \\pm 41.6$ s (\\citealt{151006A_refined_BAT}). The XRT started observations nearly \n$48.6$ s after the BAT trigger and located an X-ray source at RA = $147.43 \\,\\rm d$ and DEC = $70.5 \\,\\rm d$. \nThe XRT observed in Window Timing (WT) mode during 55.3--570.8\\,s, and in photon counting mode \nafterwards until 114\\,ks (both the time counted from the BAT trigger time). The long term XRT light curve \nis best fitted with a double-broken power law with slopes of $-0.5_{-2}^{+0.2}$ until 91\\,s, $-2.1\\pm0.4$ until\n134\\,s and $-1.39\\pm0.02$ afterwards (XRT repository; \\citealt{Evansetal_2009}). In the current analysis we have included  \nthe XRT data only until $\\sim 600\\, \\rm s$, i.e the part with WT mode observation and nearly coincident \nwith the observed BAT emission.\\footnote{For a full light curve of the XRT observations, please refer to the \nonline repository in the link \\url{http:\/\/www.swift.ac.uk\/xrt_curves\/00657750\/}}\n\nGRB 151006A was also the first GRB detected by the \\emph{Astrosat} CZTI, on it's first day of operation\n(\\citealt{151006A_gcn_czti, Raoetal_2016}). \nBoth CZTI and Veto detector light curves also show a single FRED \nlike pulse in both the energy ranges 50--200\\,keV  and 100 - 500 keV, see Figure 2 in \\cite{Raoetal_2016}. \n\nFor the spectral study with the {\\it Fermi}\/GBM, we choose three NaI detectors having the highest count rate, namely \nn0, n1 and n3, where the numbering of the NaI detectors follows the usual convention, i.e., `nx' with $x=0-11$. \nAs the number of all the NaI detectors are within $x\\le5$, we choose the BGO number b0, where `by' \ndenotes a BGO detector with $y=0-1$. \nWe then use Fermi Burst Analysis GUI v 02-01-00p1 \n({\\sc gtburst}\\footnote{\\url {http:\/\/fermi.gsfc.nasa.gov\/ssc\/data\/analysis\/scitools\/gtburst.html}}) \nto extract the spectrum. As the n3 has the \nhighest count rate among the chosen NaI detectors, we use it to define the time intervals for time resolved\nspectroscopy. We apply a signal-to-noise ratio (S\/N) of 20 and find 14 time bins in the interval \n$-2.0-88.1$\\,s. The LAT LLE data is also extracted in these time bins until $\\sim 27$\\,s, following\nthe standard procedure described in \\cite{Ackermannetal_2013_LAT}. \nThe LAT P8R2$\\_$source class events, $>100$ MeV, events were selected within a $12 \\deg$\nregion centred around the {\\it Swift}~XRT position. Among the 5 detected LAT photons, only 4  \narrive during the GBM $T_{\\rm 90}$ of the burst. Thus, for the temporal analysis, \ndepending on the availability of the data, the LAT spectra is extracted in the energy \nrange, $100 \\,\\rm MeV - 1\\,\\rm GeV$, only for time intervals:  16.3 - 20.5 s (bin 8) and 37 -53 s (bin 11).\n\nThe first $88$\\,s of the {\\it Fermi}~ data is augmented by the observation with the {\\it Swift}\/BAT ($88-253$\\,s) and {\\it Swift}\/XRT \nat later times ($> 58.5$\\,s). \nThe data of the BAT was extracted following the standard procedure. The data is calibrated using the task\n{\\sc bateconvert}, followed by constructing a detector plane image using {\\sc batbinevt}. The known\nbad detectors and the noisy pixels are eliminated by {\\sc batdetmask} and {\\sc bathotpix}. It is then mask\nweighted by {\\sc batmaskwtevt}. Finally, the spectrum is extracted with {\\sc batbinevt} in a specified \ntime interval. The spectrum is corrected by  ray-tracing with {\\sc batupdatephakw} and the response \nmatrix is generated using {\\sc batdrmgen}. The XRT data was extracted using the standard tools provided \nby the UK Swift Science Data Centre (\\citealt{Evansetal_2009})\\footnote{\\url {http:\/\/www.swift.ac.uk\/burst_analyser\/}}.\nWe extract the spectrum from the WT data with a pileup and exposure map correction. \n\nAs for the spectral model, we first choose Band function (\\citealt{Bandetal_1993}), which \nis a broken power law with two photon indices, $\\alpha$ and $\\beta$ and a peak, $E_{\\rm p}$ \nin the $\\nu F_\\nu$ representation. A majority of GRB \ndata is consistent with this function (e.g., \\citealt{Gruberetal_2014, Goldstein_etal_2013}).\nAs we will show in Section~\\ref{spectral_evolution}, the evolution of the $E_{\\rm p}$ of the \nBand function with time shows a sudden jump. As GRB spectrum has been found to have multiple spectral components, this \nsudden jump can as well represent another peak in the spectrum which is not captured \nby the single peak Band function. In order to check that the spectral variation is real, we then \nuse two models, Band function + Blackbody (Band + BB), e.g., \n\\cite{Guiriecetal_2011, Axelsson_etal_2012, Guiriecetal_2013} and \nTwo blackbodies + power law (2BB + PL), e.g., \\cite{Basak_Rao_2014_MNRAS,\nBasak_Rao_2015_090618, Iyyanietal_2015}. In addition, at the later phase\nwhere the data does not allow to put a constrain on the high energy power law,\nwe use instead a blackbody + power law (BB + PL) model.\n\nThe spectral analysis is carried out in {\\sc xspec} version: 12.9.0. For the analysis involving {\\it Fermi}\/GBM \nand LAT data, PG-Statistic is used (\\citealt{Greiner_etal_2016}) and that involving {\\it Swift}\/BAT and XRT data, \n$\\chi^2$ statistic\\footnote{\\url {http:\/\/swift.gsfc.nasa.gov\/analysis\/bat_swguide_v6_3.pdf}} is used.\nAll the errors on the fit parameters are quoted at $1\\sigma$ (nominal 68\\% confidence). \n\n\n\n\\begin{table*}\\centering\n\\caption{Parameters of time resolved spectral fitting with Band model. Bin 8 and 11 have a \nsimultaneous coverage with the LAT data marked $^{(a)}$\n\\begin{tabular}{ccccccccc}\n\n\\hline\nBin \\# & Time interval (s) & $\\alpha$ &  $\\beta$ & $E_{\\rm peak}$\\,(keV)  & $N_{\\rm Band}\\,(10^{-3})$ &  PG-Stat (dof)  \\\\ \n\\hline\n\\hline \n$1$        &$-2.0$ -- $ 2.7$ & $   -0.90_{ -0.06}^{+ 0.05}$    & $   -2.7_{-0.2}^{+0.1}$   & $ 3082_{  -631}^{+ 1110}$    & $    3.4_{-0.2}^{+ 0.2}$  &  $  489.8\\,(477)$   \\\\\n$2$        &$ 2.7$ -- $ 4.5$ & $    -1.13^{+0.04}_{-0.05}$     & $   -2.4^{ +0.1}_{- 0.2}$ & $ 2325^{+3475}_{-620}$       & $ 10.9_{- 0.5}^{ +0.4}$   &  $ 457.2\\,(477)$   \\\\\n$3$        &$ 4.5$ -- $ 6.3$ & $  -1.10^{+0.11}_{-0.04}$       & $   -2.5^{ +0.2}_{- 0.2}$ & $ 921^{ +9036}_{- 416}$      & $ 12.3^{ +2.0}_{- 0.6}$   &  $  532.3\\,(477)$ \\\\\n$4$        &$ 6.3$ -- $ 8.2$ & $  -1.14^{+ 0.29}_{- 0.09}$     & $   -2.3^{+ 0.2}_{- 0.2}$ & $ 625^{+670}_{- 424}$        & $ 13.3^{+ 7.7}_{- 1.5}$   &  $ 499.2\\,(477)$ \\\\\n$5$        &$ 8.2$ -- $10.3$ & $  -1.25^{+ 0.10}_{-0.03}$      & $   -2.5^{+ 0.1}_{- 0.2}$ & $ 1292^{+ 669}_{- 596}$      & $ 10.5^{+ 1.4}_{- 0.4}$   &  $ 507.8\\,(477)$ \\\\\n$6$        &$10.3$ -- $13.2$ & $ -1.18^{+0.09}_{- 0.08}$       & $   -2.4^{ +0.1}_{- 0.2}$ & $ 582^{ +370}_{- 191}$       & $ 9.9^{+ 1.3}_{-1.1}$     & $   471.2\\,(477)$\\\\\n$7$        &$13.2$ -- $16.3$ & $  -1.16^{+ 0.11}_{-0.09}$      & $   -2.3^{+ 0.1}_{-0.1}$  & $ 411^{ +237}_{- 135}$       & $ 10.1^{ + 1.8}_{- 1.3}$  & $  529.5\\,(477)$\\\\ \n$8^{(a)}$  &$16.3$ -- $20.6$ & $  -1.21^{+ 0.13}_{-0.12}$      & $   -2.5^{+ 0.1}_{-0.2}$  & $ 492^{+ 619}_{-204}$        & $ 7.6^{+ 1.7}_{-1.3}$     & $  574.9\\,(487)$\\\\\n$9$        &$20.6$ -- $27.5$ & $  -1.08^{+0.22}_{- 0.30}$      & $   -2.2^{+ 0.1}_{-0.3}$  & $ 208^{+950}_{-86}$          & $7.3^{+3.3}_{-2.9}$       & $   612.8\\,(477)$\\\\\n$10$       &$27.5$ -- $37.1$ & $   -1.34^{+ 0.16}_{-0.10}$     & $   -1.6^{+0.1}_{-0.1}$   & $ 580^{+6619}_{- 397}$       & $3.1^{+0.8}_{-0.4}$       & $   640.5\\,(469)$\\\\\n$11^{(a)}$ &$37.1$ -- $53.2$ & $   -1.34^{+ 0.06}_{- 0.06}$    & $   -3.1^{+0.4}_{- 0.3}$  & $ 8318^{+7462}_{-4801}$      & $1.6^{+0.1}_{-0.1}$       & $   747.4\\,(479)$\\\\ \n\n\\hline\n\\end{tabular}\n\n\\label{t_band}\n\\end{table*}\n\n\n\n\n\n\\subsection{Effective area correction:} \nThe different detectors used for the spectral analysis are expected to have some differences\nin the calibration. We multiply a constant factor for each detector used for a spectral fitting\nfor the effective area correction (EAC). However, as each time-resolved data has a limited number \nof counts, the EAC constant may not be well constrained. Hence, we adopt the following procedure.\n\nFirst, note that it is sufficient to determine the correction factor of all the detectors relative\nto one of the detectors. Hence, we freeze the EAC constant value corresponding to the detector having \nthe highest count rate to 1, while that of the other detectors are made free. We then load the time-resolved \ndata of all the detectors in all the time bins simultaneously. Then the EAC constant parameter of \nthe individual detectors are linked throughout all the time bins. For example, let us assume that \nwe have three detectors and four time bins, with detector 1 having the highest count rate. \nWe load $3\\times4=12$ data groups simultaneously. Group 1--3 correspond to time bin 1, group 4--6 \ncorrespond to time bin 2 and so on. Now, the constant parameter of group 1 is fixed to 1, while that of \ngroup 2 and 3 are free. From the next time bin onward this parameter is linked to the \ncorresponding value of time bin 1. Thus, the constant parameter of the group 4 is \nset equal to that of the group 1, that of the group 5 is set equal to that of the group 2,\nand so on. Note that this procedure is allowed as the EAC of the detectors cannot vary with time.\nOn the other hand, linking this parameter throughout all the time bins enable us to determine \nthe value with much higher accuracy than that obtained by individual fits.\n\n\nIn the present study, as the n3 detector has the highest count rate, we freeze the EAC constant \nof this to 1. As the LAT-LLE data is available until bin 9, we use the procedure until this time \nbin. For the various models we use, we find that the EAC constant of all the detectors of GBM give quite \nsimilar values. These are --- n3: 1 (fixed), n0: $0.95\\pm0.03$, n1: $0.93\\pm0.03$, b0: $1.0\\pm0.1$.\nThe EAC constant corresponding to the LAT-LLE was found to widely differ between models. The value \nis sometimes unrealistically low ($<0.3$). Based on the current understanding of the different \ndetectors of the {\\it Fermi},~ the EAC constant factor is not expected to differ by more than $30\\%$.\nWe find that the Band + BB gives the most realistic value of this factor \nin the range $0.40-1.02$. Note that due to much less number of spectral bins, the \nconstant factor is not well constrained even after using the above mentioned procedure. As the value is \nconsistent with having no correction, we freeze the EAC constant factor corresponding to the LAT-LLE \nand the LAT data ($>100$\\,MeV) \nto 1 for all spectral fitting. For the XRT analysis, we find the constant factor in \nthe range 0.8--1.2, and hence, we freeze it to 1. We do not apply any cross-calibration between the XRT \nand the BAT.\n\n\\section{Results}\n\\label{results}\n\n\\subsection{Spectral evolution during the prompt emission}\n\\label{spectral_evolution}\n\n\\begin{figure}\\centering\n{\n\\includegraphics[width=\\columnwidth]{band_parm-eps-converted-to.pdf} \n}\n\\caption{Evolution of the parameters of the Band model fitted to the time-resolved \\emph{Fermi}\/GBM,\nLAT and XRT data. For Bin 12--14, we show the parameters of the power law model fits in orange \nsymbols. Top to bottom --- Panel 1: $E_{\\rm p}$; Panel 2: Photon index $\\alpha$ \n(red filled boxes), $\\beta$ (black filled circles) of the Band function, and the \npower law index, $\\Gamma$ (orange filled boxes); \nPanel 3: Energy flux in units of $10^{-7}$\\,erg\\,cm$^{-2}$\\,s$^{-1}$. The light curve in energy range,\n8 keV -- 100 keV, is shown in gray in Panel 1.\nThe detection of the first LAT photon at 17.5\\,s is marked by a vertical dashed line in Panel 1.\nIn Panel 2, the synchrotron fast cooling photon index of 3\/2 and the slow cooling photon index of 2\/3 are marked by \nhorizontal dashed lines.\n}\n\\label{band_parm}\n\\end{figure}\n\n\n\n\\begin{figure}\\centering\n{\n\\includegraphics[width=\\columnwidth]{wide_spectrum-eps-converted-to.pdf} \n}\n\\caption{Wide band spectrum of Bin 8 fitted with the Band function. Markers\n(and colours) used for different detectors are shown in the legend.\n}\n\\label{wide_spectrum}\n\\end{figure}\n\n\n\\begin{figure}\\centering\n{\n\\includegraphics[width=\\columnwidth]{babb_parm-eps-converted-to.pdf} \n}\n\\caption{Evolution of the parameters of the Band + BB model fitted to the time-resolved \\emph{Fermi}\/GBM,\nLAT and XRT data. For Bin 10 --14, the parameters of the BB + PL model fits are shown in orange \nsymbols. From top to bottom --- Panel 1: $E_{\\rm p}$ of the Band (black filled circles) and $kT$ \nof the blackbody (red open circles); Panel 2: Photon index $-\\alpha$ (red filled boxes), \n$-\\beta$ (black filled circles) of the Band function, and the power law index, $\\Gamma$ (orange filled boxes);\nPanel 3: Energy flux in units of $10^{-7}$\\,erg\\,cm$^{-2}$\\,s$^{-1}$\nfor Band (black filled circles), BB (red open circles) and power law (orange filled circles);\nPanel 4: $\\cal{R}$ in units of $10^{-21}$. The light curve in energy range, 8 keV - 100 keV, is shown in grey in Panel 1.\nThe detection of the first LAT photon at 17.5\\,s is marked by a vertical dashed line in Panel 1.\nIn Panel 2, the synchrotron fast cooling photon index of 3\/2 and the slow cooling photon index of 2\/3 are marked by \nhorizontal dashed lines.}\n\\label{babb_parm}\n\\end{figure}\n\n\\begin{figure}\\centering\n{\n\\includegraphics[width=\\columnwidth]{2bbpl_parm-eps-converted-to.pdf} \n}\n\\caption{Evolution of the parameters of the 2BB + PL and BB + PL model fitted to the time-resolved \\emph{Fermi}\/GBM,\nLAT and XRT data are shown. The parameters of the high energy blackbody, low energy blackbody \nand the power law component of the 2BB + PL model\nare shown in black filled circles, red open circles and violet open boxes respectively.\nThe parameters of  BB + PL model fits are shown in symbols of orange colour.\nFrom top to bottom --- Panel 1: Temperature of the blackbodies; Panel 2: Power law photon index, $\\Gamma$; \nPanel 3: Energy flux of each component in units of $10^{-7}$\\,erg\\,cm$^{-2}$\\,s$^{-1}$;\nPanel 4: $\\cal{R}$ in units of $10^{-21}$. The light curve in energy range, 8 keV - 100 keV, is shown in grey in Panel 1. \nThe detection of the first LAT photon at 17.5\\,s is marked by a vertical dashed line in Panel 1.\n}\n\\label{2bbpl_parm}\n\\end{figure}\n\n \nThe results of the Band fits are shown in Table~\\ref{t_band} and Fig.~\\ref{band_parm}.\nAn example of Band fit is shown for Bin 8, where the wide band data including that of \nthe LAT is available, see Fig.~\\ref{wide_spectrum}.\nThe parameters could be constrained until Bin 11, and afterwards the Band turns to be a \nsingle power law. Note that the $E_{\\rm p}$ evolution in Panel 1 of Fig.~\\ref{band_parm} shows a \nhard-to-soft (HTS) evolution until Bin 9 and then the value starts to increase and reaches a few MeV. \nWhen the spectral evolution is compared with the 8 keV --100\\,keV light curve, shown in the background, the\nevolution does not seem to show an intensity-tracking (IT) trend either. Note from Fig.~\\ref{lc} that none \nof the light curves in any energy band show any new pulse at this phase. \nHowever, interestingly, we note that the first LAT photon ($ > 100\\, \\rm MeV$) arrives at 17.5\\,s,\nwhich is marked by a dashed line in both Fig.~\\ref{lc} and Fig.~\\ref{band_parm}. \nThe observed change in the spectral peak of the Band function\ntakes place during this time. This thereby \nsuggests either an onset of a second hard pulse related to the prompt emission \nor the beginning of the afterglow phase (see Section~\\ref{second_pulse} and \nSection~\\ref{disc} for a detailed discussion).\n\n\n\nIn the second panel of Fig.~\\ref{band_parm}, we show the evolution of the photon indices of the Band \nfunction as well as that of the power law. Negative values of $\\alpha$ and $\\beta$\nare shown in order to match the convention of the photon index $\\Gamma$ of the power law model i.e., \n$N(E)\\propto E^{-\\Gamma}$. \nThe $\\alpha$ values are found to be softer than the line of death of synchrotron emission (\\citealt{Preeceetal_1998})\ni.e $\\alpha = -0.67$, corresponding to slow cooling synchrotron emission,\nthroughout the burst duration. In accordance to what is typically observed \n(\\citealt{Kanekoetal_2006}), $\\alpha$ is found to get softer with time, tending to values, \n$\\alpha = -1.5$, consistent with fast cooling synchrotron emission (marked as pink horizontal dash\nline in Fig.~\\ref{band_parm} ).  The value of $\\beta$ also decreases gradually over time. \nAs a result of which, at later times, it becomes difficult to constrain the spectral peak and \nspectrum is then modelled by a simple power law. The power law index, $\\Gamma$ has values equal\nto $1.5$. Such hard values of $\\Gamma < 2$ suggest either a spectral peak or cutoff to lie beyond \nthe observed energy window, e.g \\cite{Gonzalezetal_2003}.  \n\n\nIn the third panel we show the evolution of bolometric energy flux in 0.1\\,keV -- 100\\,GeV.\nWe however note that the bolometric \nvalue inherently assumes that the same model can be extended in both lower and higher energies. While an observed \nflux would not have such assumption, it can underestimate the powerlaw flux. Also, as we have used different \ndetectors at different phases of the GRB, the bolometric flux automatically provides a uniform band for all\nobservation. The flux at any desired energy band can be easily calculated using the parameters given in the tables.\nThe Band flux smoothly decreases until bin 9.\nNote that the power law model as well as in case of the Band function (bin10), where the Band nearly tends \nto be a power law ($\\beta$ is nearly equal to $\\alpha$), show an increased flux in comparison to the \notherwise smooth evolution. This is due to the fact that in other bins, the Band function has a \nsteeper slope at higher energies than the power law which does not have a break. \nConsequently, in these bins the fluxes are over estimated. \n\n\nThe apparent jump in $E_{\\rm p}$ evolution as found above calls for a detailed study. This is solely a spectral \nvariation as the underlying lightcurve does not show any change. It is known that GRB spectrum can \nhave multiple components with two peaks e.g., Band + BB, or 2BB + PL. If one of the \ncomponents is dominant, it is possible that our single component Band function picks up that one\nleading to an artificial $E_{\\rm p}$ variation. Hence, in order to check that the variation is \nphysical we use these two models. The corresponding\nspectral evolutions are shown in Fig.~\\ref{babb_parm} for Band + BB and Fig.~\\ref{2bbpl_parm}\nfor 2BB + PL model. Whenever the curvature at high energies cannot be constrained,\nwe use the simpler BB + PL model, which these two models would converge to. \nFor the parameters of all the fits, see Appendix~\\ref{table_spectral}.\nFor bins 12--14, we include simultaneous XRT observation as well. \nAs soft X-rays suffer from absorption at the source and the Galaxy, we include two absorption \nterms ({\\sc TBabs} in {\\sc Xspec}) in the model. The equivalent hydrogen column density ($n_{\\rm H}$) of the \nGalactic term is fixed to $7.74\\times10^{20}$\\,atoms cm$^{-2}$. Then we link the $n_{\\rm H}$ of the \nsource absorption term between the time bins. We obtain $n_{\\rm H}=(5.1\\pm0.6)\\times10^{21}$\\,atoms cm$^{-2}$.\nFor subsequent fitting with the XRT data, we freeze the source  $n_{\\rm H}$ at this value as this cannot \nevolve with time and the linking of the parameter gives good confidence on the obtained value.\n\nFrom the upper panels of Fig.~\\ref{babb_parm} and Fig.~\\ref{2bbpl_parm} we see a very similar \nvariation of the spectral peak as before. This indeed shows that the sudden jump in the peak \nenergy is not an artefact of the adopted model. \nThe next two panels of the Figures are same as \nFig.~\\ref{band_parm}. In the fourth panel, we also show the \nevolution of the parameter $\\cal{R}$$\\equiv(F_{\\rm BB}\/\\sigma T^4)^{1\/2}$ of the blackbody component. \n${\\cal{R}}$ represents the effective transverse size of the emitting region (i.e photosphere) \nprovided the bulk Lorentz factor of the outflow, $\\Gamma_{\\rm B} \\gg 1\/\\theta_{\\rm j}$ where\n$\\theta_{\\rm j}$ is the jet opening angle (\\citealt{Ryde_Pe'er_2009}).\nFor Band + BB note that $\\cal{R}$ does not show a linear increase in contrast to what is typically observed\n(\\citealt{Ryde_Pe'er_2009,Ryde_2004,Ryde_2005}), instead only exhibits an overall increment throughout.\nOn the other hand, the variation of $\\cal{R}$ for both the thermal components of 2BB + PL \nmodel exhibit a similar jump corresponding to the evolution of $E_{\\rm p}$.\n\n\n\n\n\n\n\\begin{figure}\\centering\n{\n\\includegraphics[width=\\columnwidth]{2bbpl_late_evolution-eps-converted-to.pdf} \n}\n\\caption{Long time evolution of the parameters of the 2BB + PL model are shown. \nThe symbols used are the same as in Fig.~\\ref{2bbpl_parm}. The data after 88\\,s \ncorrespond to Table~\\ref{t_late_2bbpl}. The 0.3--10 keV XRT flux\n(erg\\,cm$^{-2}$\\,s$^{-1}$) is shown in the background of Panel 1\nand the corresponding vertical scale is shown on the right.\nThe value of $\\cal{R}$ of the lower-temperature blackbody is not shown for \nthe last time interval, as due to a very low temperature, we obtained an \nunrealistically high value.\n}\n\\label{2bbpl_late_evolution}\n\\end{figure}\n\n\\begin{figure}\\centering\n{\n\\includegraphics[width=\\columnwidth]{babb_late_evolution-eps-converted-to.pdf} \n}\n\\caption{Long time evolution of the parameters of the Band+BB model (Bin 1--9) and \nBB+PL model (Bin 10--18) are shown.\nThe symbols used are the same as in Fig.~\\ref{babb_parm}. \nThe 0.3--10 keV XRT flux\n(erg\\,cm$^{-2}$\\,s$^{-1}$) is shown in the background of Panel 1\nand the corresponding vertical scale is shown on the right.\n}\n\\label{babb_late_evolution}\n\\end{figure}\n\n\n\n\n\\subsection{Spectral evolution at late times ($> 88 \\,\\rm s$)}\n\\label{late_phase}\n\n\n\n\n\n\\begin{table*}\\centering\n\\caption{Parameters of time resolved spectral fitting of the joint BAT and XRT data at late times.\nWe also show the $\\chi^2$ of the models, including that of the Band function fits.\n\\begin{tabular}{cccccccccc}\n\n\\hline\nBin \\# & Interval (s) & Model & $kT_{\\rm h}$\\,(keV) &  $N_{\\rm h}$\\,($10^{-1}$) & $kT_{\\rm l}$ or $kT$\\,(keV) & $N_{\\rm l}$\\,($10^{-1}$) & $\\Gamma$ & $N_{\\Gamma}$\\,($10^{-1}$) & $\\chi^2$\\,(dof) \\\\\n\\hline\n\\hline\n$15$ &$88.2-94.2$     & PL    &                             &                             &                             &                          & $  1.39_{ -0.05}^{+  0.05}$ & $  5.4_{ -0.4}^{+  0.5}$ & $ 113.4\\,(75)$ \\\\\n     &                & BBPL  &                             &                             & $  12_{ -2}^{+  2}$         & $  2.1_{ -0.3}^{+  0.3}$ & $  1.97_{ -0.12}^{+  0.13}$ & $  8.4_{ -0.8}^{+  0.8}$ & $  86.9\\,(73)$ \\\\\n     &                & 2BBPL & $  38_{ -13}^{+  52}$ & $  3.8_{ -1.4}^{+ 13}$            & $  6_{ -1}^{+  1}$          & $  1.4_{ -0.3}^{+  0.3}$ & $  2.21_{ -0.16}^{+  0.17}$ & $  8.9_{ -0.8}^{+  0.8}$ & $  70.8\\,(71)$ \\\\\n&&Band&&&&&&&103.1 (73)\\\\\n     \\hline\n$16$ &$94.2-104.2$    & PL    &                             &                             &                             &                          & $  1.40_{ -0.04}^{+  0.04}$ & $  4.6_{ -0.3}^{+  0.3}$ & $  88.9\\,(75)$ \\\\\n     &                & BBPL  &                             &                             & $  11_{ -2}^{+  2}$         & $  1.1_{ -0.3}^{+  0.3}$ & $  1.64_{ -0.10}^{+  0.11}$ & $  5.4_{ -0.5}^{+  0.5}$ & $  77.8\\,(73)$ \\\\\n     &                & 2BBPL & $  17_{  -3}^{+   4}$ & $  1.6_{ -0.3}^{+  0.3}$          & $  4_{ -1}^{+  1}$          & $  0.8_{ -0.2}^{+  0.2}$ & $  2.14_{ -0.22}^{+  0.23}$ & $  6.2_{ -0.5}^{+  0.5}$ & $  70.4\\,(71)$ \\\\\n&&Band&&&&&&&82.8 (73)\\\\\n\\hline\n$17$ &$104.2-137.2$   & PL    &                             &                             &                             &                          & $  1.43_{ -0.03}^{+  0.04}$ & $  3.0_{ -0.1}^{+  0.1}$ & $  89.5\\,(75)$ \\\\\n     &                & BBPL  &                             &                             & $  14_{ -5}^{+  8}$         & $  0.4_{ -0.2}^{+  0.2}$ & $  1.55_{ -0.07}^{+  0.07}$ & $  3.2_{ -0.2}^{+  0.2}$ & $  84.9\\,(73)$ \\\\\n     &                & 2BBPL & $  25_{  -5}^{+  10}$ & $  1.1_{ -0.3}^{+  0.5}$          & $  4.3_{ -0.4}^{+  0.6}$    & $  0.5_{ -0.1}^{+  0.1}$ & $  1.93_{ -0.14}^{+  0.15}$ & $  3.6_{ -0.2}^{+  0.2}$ & $  72.8\\,(71)$ \\\\\n&&Band&&&&&&&77.8 (73)\\\\\n\\hline\n$18$ &$137.2-253.2$   & PL    &                             &                             &                             &                          & $  1.38_{ -0.04}^{+  0.04}$ & $  0.92_{ -0.04}^{+  0.04}$ & $  96.8\\,(75)$ \\\\\n     &                & BBPL  &                             &                             & $  7_{ -1}^{+  1}$          & $  0.3_{ -0.1}^{+  0.1}$ & $  1.59_{ -0.08}^{+  0.09}$ & $  1.00_{ -0.06}^{+  0.06}$ & $  75.8\\,(73)$ \\\\\n     &                & 2BBPL & $   7_{  -1}^{+   1}$ & $  0.2_{ -0.1}^{+  0.1}$          & $  0.07_{ -0.01}^{+  0.02}$ & $  0.2_{ -0.1}^{+  0.3}$ & $  1.51_{ -0.07}^{+  0.08}$ & $  0.93_{ -0.06}^{+  0.06}$ & $  65.8\\,(71)$ \\\\\n&&Band&&&&&&& $96.2 (73)$\\\\\n\\hline\n$19$ &$253.2-573.2$   & PL    &                             &                             &                             &                             & $  1.50_{ -0.06}^{+  0.06}$ & $  0.03_{ -0.02}^{+  0.02}$ & $  14.3\\,(17)$ \\\\\n\\hline\n\n\n\\end{tabular}\n\n\\label{t_late_2bbpl}\n\\end{table*}\n\nThe BAT and XRT spectrum at late times are extracted \nwhen the S\/N of the {\\it Fermi}~ does not allow any further time division. The XRT data in WT mode\nand the BAT data are available until 570\\,s and $\\sim250$\\,s, respectively. Note that the \nXRT light curve has two breaks one at 91\\,s and another at 134\\,s, both counted with respect \nto the BAT trigger time. Hence, in choosing the time intervals, we respect these break times \nand choose roughly logarithmic bins i.e., the interval length increases roughly as a\ngeometric progression. We obtain 5 bins: 85--91\\,s, 91--101\\,s, 101--134\\,s, 134--250\\,s\nand 250--570\\,s (from BAT trigger time). The simultaneous BAT data is extracted in the first \nfour intervals.\n\nThe time-resolved data is analysed with a power law (PL), BB + PL, 2BB + PL and Band function. As \nbefore, two absorption terms --- the Galactic $n_{\\rm H}$ is set to \n$7.74\\times10^{20}$\\,atoms cm$^{-2}$ and the source $n_{\\rm H}$ to $5.1\\times10^{21}$\\,atoms cm$^{-2}$, are used.\nThe result of the fits are shown in Table~\\ref{t_late_2bbpl}. We find that the PL \ndoes not give a good fit (see Table~\\ref{t_late_2bbpl}) and shows large residual. The spectra are much better fitted with \nthe BB+ PL model and even better with the 2BB + PL model. This shows that there is\na curvature in the spectrum. Hence, this phase most probably corresponds to the prompt \nemission phase. We found that the Band function shows an equally bad fit as the PL model \nand the $E_{\\rm p}$ could not be constrainred. The 2BB + PL model seems to be the best fit at the \nlate time. For the last bin, as the BAT data is no more available we could not constrain the \nparameters of more complicated models other than the PL in the limited bandwidth of the XRT.\nHence, only the results of PL fit is reported for this bin. \n\nIn Fig.~\\ref{2bbpl_late_evolution}, we show the evolution of the parameters of the \n2BB + PL model taking all the observations together. The panels are same as in Fig.~\\ref{2bbpl_parm}\nthat refers to the 2BB + PL model fitted at early times. In Fig.~\\ref{babb_late_evolution}, \nsimilar plot is shown for the Band + BB model. In both cases, the evolution at the late phase \nis consistent with the previous evolution. The late time spectral evolution together with a \nsignificantly better reduced $\\chi^2$ in the high resolution data of the XRT \nsuggests that the 2BB + PL model probably most consistently captures the \noverall spectral evolution of the burst. \n\n\n\n\n\\subsection{Evidence for a second hard pulse}\\label{second_pulse}\n\n\\begin{figure}\\centering\n{\n\\includegraphics[width=\\columnwidth]{LC_HR-eps-converted-to.pdf} \n}\n\\caption{\\emph{Upper panel:} Bayesian blocks shown on the light curve of combined NaI 0, 1 and 3 in 8--900\\,keV.\nThe blocks are -1--1\\,s, 1--10\\,s, 10--15\\,s and 15--24\\,s. clearly, a new block starts near the time of first \ndetected LAT photon at $\\sim 18$\\,s. \\emph{Lower panel:} Hardness ratio defined as the ratio of count between \n500\\,keV--5\\,MeV and 200\\,keV--400\\,keV band of BGO 0. A sudden increment of hardness is apparent at $\\sim 18$\\,s.\n}\n\\label{lc_hr}\n\\end{figure}\n\n\nIn case of GRBs with multiple but separable pulses, it is frequently observed that the first pulse shows \na HTS evolution, followed by a jump in the peak energy during the onset of a second pulse,\nand then again showing a HTS evolution in the falling part of that pulse (see e.g., \n\\citealt{Ghirlandaetal_2010, Luetal_2012}). \nA very similar behaviour is seen for GRB 151006A, though the presence of a second pulse is not \nseen in the lightcurve. The late time spectral evolution being consistently HTS since the time of sudden jump\n($\\gtrsim 18$\\,s, see Section~\\ref{late_phase}) indicates that the second phase is probably a second pulse \nof the prompt emission hidden in the data. This is not readily evident probably because it is expected to be a hard pulse, \nwhere the signal is weak. In order to have a better look at the phenomenon we perform the following analysis. \n\nWe first construct the Bayesian blocks for the combined count rate data of NaI 0, 1 and 3 in 8--900\\,keV\n(see Fig.~\\ref{lc_hr}, upper panel). The Bayesian block approach finds the best possible way to represent a \ntime-series data as a series of blocks or segments such that the signal underlying each block is \nconstant within the observational error (\\citealt{scargel2013}). We use the dynamical programing \nalgorithm of \\cite{scargel2013} to construct these blocks which are then over-plotted on the \ncount rate lightcurve in Fig.~\\ref{lc_hr}, upper panel. During the interval shown in the Figure, \nwe find the Bayesian blocks as -1--1\\,s, 1--10\\,s, 10--15\\,s and 15--24\\,s. With only the NaI 3 \ndetector these blocks are -1--1\\,s, 1--14\\,s and 14-24\\,s. We note that a new block starts \nat $\\sim14-15$\\,s which shows that the count rate flux of this bin is statistically different from the \nBayesian blocks on either side on the time axis. But, as the flux level of the consecutive blocks \naround this block decreases monotonically, there is no evidence for a new pulse in the data. \nHowever, we note from Table~\\ref{t_bandbb} that the peak energy at the transition time reaches a very \nhigh value $\\sim 5$\\,MeV, and hence, it is unlikely that such a change will give rise to any\nsignificant pulse profile in the NaI detectors. On the other hand, the flux level of BGO \ndetector is quite poor to carry out such analysis. \n\nGiven the high value of the peak energy, and no pulse profile with Bayesian blocks in the NaI detector,\nwe are left with only one possibility that the pulse is hidden in the BGO data and we investigate \nthat possibility. Before proceeding, we note that there are two competing factors here:\na lightcurve constructed in a wide energy \nband will smear out the small variations that we expect at the high energies. On the other hand, a \nlightcurve in a limited bandwidth suffers from the poor statistics of photon count, more so \nfor the BGO detector and that too at a time when the photon flux is already low. This is why we used \nthe full band of the NaI detector for the Bayesian block analysis in the first place.\nA more robust way to investigate it further is to study the evolution of hardness ratio (HR)\nrather than the count rate lightcurve in a limited band. As the HR is a ratio of count between \ntwo energy bands, the small changes in the photon count subjected to the change in spectral \npeak will be amplified. More importantly, if there is a smooth pulse presumably a hard one, which is \nnot seen in the lightcurve otherwise, the HR should track that pulse profile. We use the BGO \ndetector (the BGO 0) rather than the NaI since the peak energy reaches high values covered by the BGO energy band.\nOn the other hand, we cannot use the LAT LLE data as it does not provide the low energy band required for this analysis.\nWe choose the hard band as $H=\\rm 500\\,keV-5\\,MeV$ and the soft band as $S=\\rm 200\\,keV-400\\,keV$,\nand then the Hardness Ratio $=H\/S$. \nWe then stick to the same definition throughout.\nThe evolution is shown in Fig.~\\ref{lc_hr}, lower panel.\nThe first phase shows a HTS evolution starting with a hardness ratio of $\\sim1$, \nreaching down to a value close to 0 until $\\sim15$\\,s and then shows a sudden jump again \nreaching a hardness ratio of $\\sim1$, followed by another HTS evolution. The errors \nin the hardness ratio are larger at the later phase due to the lower flux level, but the evidence \nof the jump in hardness lightcurve, and a smooth pulse profile afterwards is very clear. \nThis changeover time remarkably coincides with the onset of the unusual spectral evolution.\nThis is exactly when the second phase of emission begins \nand the first LAT photon ($>100$\\,MeV) is detected. This analysis, combined with the fact that the \nsecond phase shows a very similar HTS evolution as seen in other GRBs, points towards \nthe fact that this new energy injection is due to a second hard pulse of the prompt emission \nitself.\n\n\n\n\\begin{figure}\\centering\n{\n\\includegraphics[width=\\columnwidth]{pol-eps-converted-to.pdf} \n}\n\\caption{Modulation curves for GRB 151006A in two phases: 0--16\\,s (upper panel) and \n16--33\\,s (lower panel). The modulation curve is obtained\nafter geometrically correcting the raw azimuthal angle histogram by \nnormalizing with respect to a simulated unpolarized distribution for a \nsimilar spectra and off-axis detection angle (\\citealt{Raoetal_2016}). The blue solid line\nis the $\\cos{\\phi}^2$ fit to the modulation curve. The fitted modulation\nfactor, polarization fraction ($\\Pi$) and polarization angle (P. A.) are \nshown in the inset of the respective panels, with errors \nestimated at 68\\% confidence level (see text for details).\n}\n\\label{pol}\n\\end{figure}\n\n\\subsection{Polarization measurement -- further evidence for the second pulse of prompt emission}\n\nAnother piece of information about these two emission phases can be obtained from \npolarization measurement. If the second emission phase is a hard pulse, and we attribute \npolarization to the non thermal processes that produce harder spectrum, then we expect to have \nan enhancement of the degree of polarization in the second phase.\nCZTI being a pixelated detector and having a significant Compton scattering \nprobability at energies beyond 100 keV, essentially works as a \nCompton polarimeter at \nthese energies. The double pixel events within the photon tagging time\nwindow of 40 $\\mu$s which satisfy the Compton kinematics are identified\nas valid Compton events. The validity of the Compton event selection \nand the polarization measurement capability of CZTI have been established\nby detailed simulation and experimental studies during ground calibration\nof CZTI (\\citealt{chattopadhyay14,Vadawale_etal_2015}). The first onboard validation \nof CZTI polarimetry was obtained with the detection of polarization of\nGRB 160131A (\\citealt{vadawale16}) and GRB 151006A (\\citealt{Raoetal_2016}). GRB151006A, though\nis moderately bright for X-ray polarization measurement, a hint of \npolarization is seen with an estimated polarization degree $> 90\\%$ with\na detection significance of 1.5$\\sigma$ (68$\\%$ confidence level with \n1 parameter of interest). Since our spectroscopic analysis shows that there\ncould be two distinct phases of emission, we tried to explore polarization\nmeasurement in these two sectors using the CZTI data. Fig.~\\ref{pol} shows the\nmodulation curves for these two time bins in 100 $-$ 350 keV, where the blue\nsolid line is the sinusoidal fit to the observed distribution of \nazimuthal angle of scattering.\n\n\nIn order to estimate the polarization fractions, we did detailed \nGeant 4 simulations (\\citealt{agostinelli03}) using \\emph{Astrosat} mass model \nwhich includes all the instruments of \\emph{Astrosat} along with the complete satellite structure. We employ\nthe fitted spectroscopic models to simulate the energy distribution of the\nincident photons in Geant 4 to obtain the modulation factors for \n100$\\%$ polarized beam ($\\mu_{100}$) which are then used to estimate the polarization fractions\n($P = \\mu\/\\mu_{100}$) in two phases, 0--16\\,s and 16--32\\,s. Estimated polarization\nfractions are 77$\\%$ and 94$\\%$ with $1\\sigma$ detection significance\nat polarization angles 334$^\\circ$ and 325$^\\circ$ respectively in these \ntwo phases. The detection significance of polarization\nhas reduced significantly compared to the initial report of\npolarization for GRB 151006A (\\citealt{Raoetal_2016}) which is\nexpected due to reduced number of events in the individual phases. Though\nfraction and angle of polarization are poorly constrained in both the time \nsectors, the measured values do not show any decrement of polarization.\nIn the multi-component models with thermal and non thermal parts, it is \nsuggested that the thermal emission dominates in the first part while the \nnon thermal processes become important at the later stage (e.g., \\citealt{Gonzalezetal_2003}).\nAttributing the polarization to the non thermal processes,\nthe above evolution is consistent with the expected behaviour for such models. \nHence, there is a hint of multi-component spectra in the data.\n\nIt is important to note that the polarization degree in the afterglow phase found \nby optical measurements so far show a pretty low value ($\\lesssim10\\%$) and it is \nexpected to reduce further as the afterglow proceeds,\nsee e.g., \\cite{greiner2003, mundell2007}. This appears to be due to the fact that \nthe magnetic field during the afterglow phase is mostly generated by turbulence \nand therefore has a random orientation, having only a small coherence length. This is in contrast \nwith the prompt emission phase, where measurements show a high degree of polarization ($\\sim40-80\\%$) which \ncan be achieved by an ordered magnetic field, see \\cite{Waxman2003}. Though a different \nmechanism is possible to achieve a high degree of polarization of the prompt emission,\nthe important point is that such high values are not seen in the afterglow phase.\nOur measurement is consistent with a high value, and though the statistics is \npoor the measured values in the two phase do not contradict the interpretation\nthat we are most probably observing the prompt emission extending out to the second phase. Hence, the \ndata is indicative of a second hard pulse rather than the onset of an afterglow phase.\n\n\\subsection{Comparison between the models}\n\n\\begin{figure}\\centering\n{\\vspace{-0.1in}\n\\includegraphics[width=\\columnwidth]{pgstat_dof_allmodels-eps-converted-to.pdf} \n}\n\\caption{The PGStat\/dof of all the models are shown.\n}\n\\label{pgstat}\n\\end{figure}\n\n\n\\begin{figure}\\centering\n{\n\\includegraphics[width=\\columnwidth]{late_spectrum-eps-converted-to.pdf} \n}\n\\caption{Spectral fitting of the joint BAT and XRT data with PL, BB + PL, Band and 2BB + PL models \nfor the late time bin 15. \nThe data and residual are shown by grey filled circles. The high energy blackbody, \nlow energy blackbody (or the blackbody for the BB + PL model), the power law and\nthe total model, are shown by red dot-dashed line, orange dashed line, violet dotted\nline and thick black line, respectively.\n}\n\\label{late_spectrum}\n\\end{figure}\n\nWe then compare the different models based on their goodness of fit.\nThe PG-Stat\/dof of all the models in different time bins are shown in Fig.~\\ref{pgstat}.\nWe note that all the models have very similar PG-Stat. Thus, making the judgment of the \nbest model is quite indecisive. We then use Bayesian inference criteria (BIC) to see \nwhether the more complicated models are indeed required by statistics \n(e.g., \\citealt{wang2017}). The BIC \nis defined as $-2\\ln \\cal{L}$$+ k\\ln(\\nu+k)$, where $-2\\ln \\cal{L}$ is log likelihood,\n$k$ is the number of free parameters and $\\nu$ is the degrees of freedom. In Table~\\ref{bic}, \nwe show the BIC values and the preferred models.\n\n\n\\begin{table}\\centering\n\\caption{BIC values and preferred model\n\\begin{tabular}{ccccc}\n\n\\hline\n\\#  &  \\multicolumn{4}{c}{$-2\\ln \\cal{L}$ (BIC)}\\\\\n        & Band & Band+BB & 2BB+PL & Preferred \\\\\n\\hline\n\\hline\n1  & 489.8 (514.5)   & 488.1 (525.1)   & 501.7 (538.7)   & Band \\\\\n2  & 457.2 (481.9)   & 450.4 (525.1)   & 478.3 (515.3)   & Band \\\\\n3  & 532.3 (557.0)   & 517.0 (554.0)   & 527.4 (564.4)   & Band+BB \\\\\n4  & 499.2 (523.9)   & 485.5 (522.5)   & 483.8 (520.8)   & 2BB+PL \\\\\n5  & 507.8 (532.5)   & 506.7 (543.7)   & 515.8 (552.8)   & Band \\\\\n6  & 471.2 (495.9)   & 468.3 (505.3)   & 471.1 (508.1)   & Band \\\\\n7  & 529.5 (554.2)   & 528.6 (565.6)   & 534.1 (571.1)   & Band \\\\\n8  & 574.9 (599.7)   & 572.5 (609.7)   & 578.1 (615.3)   & Band \\\\\n9  & 612.8 (637.5)   & 601.9 (638.9)   & 604.2 (641.2)   & Band \\\\\n10 & 640.5 (665.1)   &                 &                 & Band \\\\\n11 & 747.4 (772.1)   &                 & 751.1 (788.2)   & Band \\\\\n\\hline\n   & BBPL            & BBCPL           & PL \\\\\n   \\hline\n12 & 1497 (1526) & 1498 (1534) & 1505 (1520) & PL \\\\\n13 & 1571 (1600) & 1566 (1602) & 1573 (1587) & PL \\\\\n14 & 1162 (1191) & 1162 (1199) & 1165 (1179) & PL \\\\\n\\hline\n   & BBPL            & 2BB+PL          & PL \\\\\n   \\hline\n15 & 86.9 (104.3)    & 70.8 (96.9)     & 113.4 (122.1)   & 2BB+PL \\\\\n16 & 77.8 (95.2)     & 70.4 (96.5)     & 88.9 (97.6)     & BBPL\/2BBPL \\\\\n17 & 84.9 (102.3)    & 72.8 (98.9)     & 89.5 (98.2)     & PL\/2BBPL \\\\\n18 & 75.8 (93.2)     & 65.8 (91.9)     & 96.8 (105.5)    & 2BB+PL \\\\ \n\\hline\n\\end{tabular}\n\\label{bic}\n\\end{table}\n\nWe note that in the initial bins where we use the {\\it Fermi}~ data, the Band function is the\npreferred model. This signifies that the data is consistent with the Band function \nand a more complicated model is not statistically required. There are only two exceptions,\nBin 3 and 4. In the former Band + BB is the preferred model, while in the later the \n2BB + PL is the preferred model. At the later phase, we see even the PL is the \npreferred model and the other complicated models are not statistically required. \nOur result clearly shows the limitations of a background dominated low resolution \ndetector. \n\nHowever, in the late phase, though we have lower count rate, the signal to noise of\nthe data is improved due to both BAT and XRT and we have good resolution of the XRT. \nWe see a preference for the 2BBPL model in the data at late phase.\nFrom Table~\\ref{t_late_2bbpl}, we also see that the Band function is not \npreferred. In fact, we could not constrain the peak energy, and it signifies that the \nspectral shape is not a Band function. \n\n\nIn Fig.~\\ref{late_spectrum}, we show the spectral fit and residuals of the PL, BB + PL, Band and 2BB + PL model fit\nfor bin 15. Note that for the PL fit the slope at lower and higher energies do not match which is \nclearly indicated by the deviation of the residual in the opposite direction of the zero line.\nFor the BBPL model, this is improved, but then we find large residual on the both sides of the \nblackbody peak. The Band function fares no better than the PL model and has a similar residual.\nAll the curvatures in the spectrum are consistently taken care by the 2BB + PL model. \n\n\n\\section{Conclusions and Discussion}\n\\label{disc}\n\n\n\n\nSingle pulse bursts have been carefully studied using time resolved spectroscopy, \nas it is generally assumed to alleviate the issue of pulse overlapping which hinders \nour understanding of spectral evolution of GRB with time. GRB 151006A  is a single\npulse burst in different energy bands and thereby is an ideal \none for time resolved spectroscopy. In general, such bursts exhibit a HTS\nspectral peak evolution. However, the current analysis of GRB 151006A \nshows an unexpected behaviour of change in trend from HTS\nevolution to increasing spectral peak with time, after a few seconds from \nthe burst trigger. Coincidently, this rise is observed after the arrival \nof LAT  photons ($> 100 \\, \\rm MeV$). Such a dramatic change in the spectral\nbehaviour for single pulse burst is observed for the first time. \nThis new injection of energy does not show any additional pulse\nin the counts light curve. Thus, this cautions us that a single \npulse need not always suggest a HTS spectral peak evolution and\nthat there may be other pulses hidden in the light curve profile.\n\n\n\nIt is important to notice that high energy emission is often delayed \nand generaly modelled with powerlaw appearing in the spectrum at late times\n(e.g., \\citealt{Gonzalezetal_2003, Ackermannetal_2013_LAT}). This can potentially \nchange the behaviour of the spectral evolution we report here. However, we have \ntested that a powerlaw with Band function is not statistically required\nfor Bin 8 and 11. This is apparent from the spectral shape in Fig.~\\ref{wide_spectrum}.\nGRB 151006A is not among the high-fluence LAT class  \n(cf. \\citealt{Ackermannetal_2013_LAT}), and for several such cases the spectrum \nis found to be well fitted with Band function only and no powerlaw is required. \n\n\n\n\nIn order to show that the $E_{\\rm p}$ variation is physical and not due \nto single peak Band function, we also tried models with two peaks i.e., Band + BB \nand 2BB + PL, and re-confirm the spectral variation.\nThe analysis also presents one of the key issues in current GRB spectral \nanalysis i.e., a preferred model cannot be decided based only on statistics.\nThough in the later part the 2BB + PL model seems to be a better fit.\nIn \\cite{Basak_Rao_2015_090618} the spectral components of this model is proposed to\noriginate in a spine-sheath jet (\\citealt{Ramirez-Ruizetal_2002, Zhangetal_2003, Zhangetal_2004},\nsee \\citealt{Iyyanietal_2015} for an alternative explanation).\nThe blackbody radiations are produced at the two photospheres.\nIn addition, photons crossing the boundary layer of the spine-sheath structure are \non an average Compton up-scattered and hence form a high energy power-law spectrum. \nFurther non thermal processes can occur due to \nsynchrotron emission at a higher radial distance. These photons are naturally delayed\nwith respect to the photons produced by thermal and non thermal processes stated above. \nHence, the second phase in this scenario is probably an onset of the delayed emission \nphase. We note here that though the second phase starts after $\\sim18$\\,s, the pulse \ncan have a much lower start time, which is hidden in the falling part of the first pulse.\nHence, an actual delay of this phase can be much smaller, though remains undetermined. \n\n\n\n\n\n\n\n\n\n\nRecently, \\cite{Moretti&Axelsson2016} reported a rise in the high energy spectral peak with time, \nfor the first LAT detected GRB 080825C, a multi-pulse burst (\\citealt{Abdo_etal_2009_080825C}), \nwhen it was re-analysed including the LLE data. \nThis points out the significance of LLE emission \ndetections which can be crucial in constraining the high energy \nspectral behaviour of GRB spectra. The interesting part of our \nobservation is that the increasing beahaviour is seen in an otherwise \nsingle pulse. The new injection of energy\nmay be associated with a second pulse of emission in the prompt\nphase which brings about a dynamical change in the behaviour of\nthe radiation process, or it may be an overlap of the onset of \nafterglow (emission from a different region) with the prompt \nemission, which then significantly dominates the high energy \nand thereby making the prompt emission indiscernible in the spectrum. \nWe note that the $\\alpha$ values of Band function to be consistent \nwith slow cooling synchrotron radiation in the beginning of the burst ($< -0.67$) \nand gets softer with time approaching the fast cooling index \n($\\alpha = -1.5$). If the new energy injection is due to a second pulse \nin the prompt emission, we may be observing a transition in the microphysical\nparameters related to the synchrotron radiation, such as strength or structure \nof magnetic fields, fraction of energy injected into electrons, radiation\nefficiency etc (\\cite{Daigne&Mochkovitch1998}), causing the transformation \nfrom slow to fast cooling of electrons, see \\cite{Iyyani_etal_2016}. \nSince this transformation is observed to be sudden, it perhaps indicate the\nonset of a second pulse with a different source parameter.\nFor the afterglow scenario,\nsuch emission has been mostly modelled using a power law\n(\\citealt{Ackermannetal_2013_LAT, Kumar2009, Ghisellini_etal_2010}).\nHowever, we note a definite spectral curvature at late times and \nthe detection of LAT emission \nis less significant and not many photons are detected.\nOur analysis with Bayesian block and evolution study of the hardness \nratio also points towards the former scenario i.e., a new hard pulse hidden in the data.\n\n\n\n\nFinally, the inability of determining the best model based on statistics \nalso summons for constraining observations like polarization. This can potentially\ncharacterize various radiation processes leading to the observed emission\nas well as in revealing the structure and strength of magnetic fields of emitting region.\nThis can in turn be an invaluable input in enhancing our understanding of \nshock physics as well as the content of GRB jets.  \nA significant polarization detection requires high photon\nstatistics and lack of this has actually prevented the CZTI instrument onboard \n\\emph{Astrosat} in constraining the polarization measurement for GRB 151006A. \nThe current measurement hints a moderately high polarization, however, with low \nsignificance, and thereby is consistent with nearly all model predictions.\nHowever, the capability demonstrated by CZTI, offers a promising era of \npolarization detections above $100 \\, \\rm keV$ and also its time dependent study, in case of brighter GRBs. \n\n\n\n\n\\section*{Acknowledgments} \nWe gratefully acknowledge the referee for comments on reshaping \nthe paper. This publication uses data from the \\emph{Astrosat} mission of the\nIndian Space Research Organisation (ISRO), archived at the\nIndian Space Science Data Centre (ISSDC). CZT-Imager is built\nby a consortium of Institutes across India including Tata Institute\nof Fundamental Research, Mumbai, Vikram Sarabhai Space\nCentre, Thiruvananthapuram, ISRO Satellite Centre, Bengaluru,\nInter University Centre for Astronomy and Astrophysics, Pune,\nPhysical Research Laboratory, Ahmedabad, Space Application\nCentre, Ahmedabad: contributions from the vast technical team\nfrom all these institutes are gratefully acknowledged.\nThis research has made use of data obtained through the\nHEASARC Online Service, provided by the NASA\/GSFC, in support of NASA High Energy\nAstrophysics Programs. RB is a stipendiary of START program of the Polish \nScience Foundation (2016) and supported by \nPolish National Science Centre grants \n2013\/08\/A\/ST9\/00795,\n2013\/10\/M\/ST9\/00729 and\n2015\/18\/A\/ST9\/00746.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}