process

Deformations of gerbes on smooth manifolds 3694.5 Twisted forms. Suppose that A is a sheaf of R-algebras on X. We will call anR-algebroid stack locally equivalent to A opC a twisted form of A.Suppose that S is twisted form of O X . Then, the substack iS is an OX -gerbe.The assignment S 7! iS extends to an equivalence between the 2-groupoid of twistedforms of O X (a subcategory of AlgStack C .X/) and the 2-groupoid of OX -gerbes.Let S be a twisted form of O X . Then for any U X, A 2 S.U / the canonicalmap O U ! End S .A/ is an isomorphism. Consequently, if U isacoverofX and Lis a trivialization of 0 S, then there is a canonical isomorphism of sheaves of algebrasO N0 U ! End 0 S .L/.Conversely, suppose that .U; A/ is a C-descent datum. If the sheaf of algebrasA is isomorphic to O N0 U then such an isomorphism is unique since the latter has nonon-trivial automorphisms. Thus, we may and will identify A with O N0 U. Hence, A 01is a line bundle on N 1 U and the convolution map A 012 is a morphism of line bundles.The stack which corresponds to .U; A/ (as in 4.3.3) is a twisted form of O X .Isomorphism classes of twisted forms of O X are classified by H 2 .XI OX /. Werecall the construction presently. Suppose that the twisted form S of O X is representedby the descent datum .U; A/. Assume in addition that the line bundle A 01 on N 1 U istrivialized. Then we can consider A 012 as an element in .N 2 UI O /. The associativitycondition implies that A 012 is a cocycle in LC 2 .UI O /. The class of this cocycle inHL2 .UI O / does not depend on the choice of trivializations of the line bundle A 01 andyields a class in H 2 .XI OX /.We can write this class using the de Rham complex for jets. We refer to Section 7.1for the notations and a brief review.The composition O ! O =C log ! O=C j 1 ! DR.J=O/ induces the mapH 2 .XI O / ! H 2 .XI DR.J=O// Š H 2 ..XI X ˝J X=O X /; r can /. Here the latterisomorphism follows from the fact that the sheaf X ˝ J X=O X is soft. We denote byŒS 2 H 2 ..XI X ˝ J X=O X /; r can / the image of the class of S in H 2 .XI O /.InLemma 7.13 (see also Lemma 7.15) we will construct an explicit representative for ŒS.5 DGLA of local cochains on matrix algebrasIn this section we define matrix algebras from a descent datum and use them to constructa cosimplicial DGLA of local cochains. We also establish the acyclicity of thiscosimplicial DGLA.5.1 Definition of matrix algebras5.1.1 Matrix entries. Suppose that .U; A/ is an R-descent datum. Let A 10 WD A 01 , where D pr 1 10 W N 1U ! N 1 U is the transposition of the factors. The pairings.pr 1 100 / .A 012 /W A 10 ˝R A 1 0 ! A 10 and .pr 1 110 / .A 012 /W A 1 1 ˝R A 10 ! A 10 ofsheaves on N 1 U endow A 10 with a structure of a A 1 1 ˝ .A1 0 /op -module.