process

372 P. Bressler, A. Gorokhovsky, R. Nest, and B. TsyganThe DGLA C .B/ loc Œ1 has additional variance not exhibited by C .B/Œ1. Namely,for f W Œp ! Œq – a morphism in – there is a natural map of DGLAf ] W C .B/ loc Œ1 ! C .f ] B/ loc Œ1 (5.6)defined as follows. Let f ij W .f ] B/] ij ! B f.i/f.j/ denote the tautological isomorphism.For each multi-index I D .i 0 ;:::;i n / 2 Œp nC1 letLet f nf I]WD ˝n 1j D0 f i j i j C1]W ˝n 1j D0 .f ] B/ ij i j C1!˝n 1iD0 B f.i j /f .i j C1 /:] WD ˚I 2† pnC1f In. The map (5.6) is defined as restriction along f] ] .Lemma 5.1. The map (5.6) is a morphism of DGLAf ] W C .B/ loc Œ1 ! C .f ] B/ loc Œ1:If follows that combinatorial restriction of local cochains induces the functorMC 2 .f ] /W MC 2 ..XI C .B/ loc Œ1/˝km R / ! MC 2 ..XI C .f ] B/ loc Œ1/˝km R /:Combinatorial restriction with respect to f induces the functorf ] W Def.B/ loc .R/ ! Def.f ] B/ loc .R/:It is clear that the following diagram commutes:Def.B/ loc .R/f ]Def.f ] B/ loc .R/MC 2 ..XI C .B/ loc Œ1/ ˝k m R / MC2 .f ]/ MC 2 ..XI C .f ] B/ loc Œ1/ ˝k m R /.5.2.3 Cosimplicial DGLA from descent datum. Suppose that .U; A/ is a descentdatum for a twisted sheaf of algebras as in 4.2.2. Then, for each p D 0;1;::: wehave the matrix algebra Mat.A/ p as defined in 5.1, and therefore the DGLA of localcochains C .Mat.A/ p / loc Œ1 defined in 5.2.1. For each morphism f W Œp ! Œq thereis a morphism of DGLAf ] W C .Mat.A/ q / loc Œ1 ! C .f ] Mat.A/ q / loc Œ1and an isomorphism of DGLAC .f ] Mat.A/ q / loc Œ1 Š f C .Mat.A/ p / loc Œ1induced by the isomorphism f W f Mat.A/ p ! f ] Mat.A/ q from the equation (5.5).These induce the morphisms of the DGLA of global sections.N q UI C .Mat.A/ q / loc Œ1/.N p UI C .Mat.A/fp / loc Œ1/]f .N q UI f C .Mat.A/ p / loc Œ1/.