process

Deformations of gerbes on smooth manifolds 3715.1.4 Refinement. Suppose that W V ! U is a morphism of covers. For p D0;1;:::there is a natural isomorphism.N p / Mat.A/ p Š Mat.A / pof sheaves of algebras on N p V. The above isomorphisms are obviously compatiblewith combinatorial restriction.5.2 Local cochains on matrix algebras5.2.1 Local cochains. A sheaf of matrix algebras is a sheaf of algebras B togetherwith a decompositionpMB Di;jD0as a sheaf of vector spaces which satisfies B ij B jk D B ik .To a matrix algebra B one can associate to the DGLA of local cochains defined asfollows. For n D 0 let C 0 .B/ loc D L B ii B D C 0 .B/. Forn>0let C n .B/ locdenote the subsheaf of C n .B/ whose stalks consist of multilinear maps D such thatfor any collection of s ik j k2 B ik j k1. D.s i1 j 1 ˝˝s in j n/ D 0 unless j k D i kC1 for all k D 1;:::;n 1,2. D.s i0 i 1 ˝ s i1 i 2 ˝˝s in 1 i n/ 2 B i0 i n.For I D .i 0 ;:::;i n / 2 Œp nC1 letB ijC I .B/ loc WD Hom k .˝n 1j D0 B i j i j C1; B i0 i n/:The restriction maps along the embeddings ˝n 1j D0 B i j i j C1,! B˝n induce an isomorphismC n .B/ loc !˚I 2Œp nC1C I .B/ loc .The sheaf C .B/ loc Œ1 is a subDGLA of C .B/Œ1 and the inclusion map is a quasiisomorphism.For a matrix algebra B on X we denote by Def.B/ loc .R/ the subgroupoid ofDef.B/.R/ with objects R-star products which respect the decomposition given by.B˝kR/ ij D B ij ˝kR and 1- and 2-morphisms defined accordingly. The compositionDef.B/ loc .R/ ! Def.B/.R/ ! MC 2 ..XI C .B/Œ1/ ˝k m R /takes values in MC 2 ..XI C .B/ loc Œ1/ ˝k m R / and establishes an isomorphism of2-groupoids Def.B/ loc .R/ Š MC 2 ..XI C .B/ loc Œ1/ ˝k m R /.5.2.2 Combinatorial restriction of local cochains. Suppose given a matrix algebraB D L qi;jD0 B ij is a sheaf of matrix k-algebras.