process

350 P. Bressler, A. Gorokhovsky, R. Nest, and B. TsyganThe DGLA g.J X / ŒS is defined as the ŒS-twist of the DGLAg DR .J X / WD .XI DR. xC .J X //Œ1/:Here, J X is the sheaf of infinite jets of functions on X, considered as a sheaf of topologicalO X -algebras with the canonical flat connection r can . The shifted normalizedHochschild complex xC .J X /Œ1 is understood to comprise locally defined O X -linearcontinuous Hochschild cochains. It is a sheaf of DGLA under the Gerstenhaber bracketand the Hochschild differential ı. The canonical flat connection on J X induces one, alsodenoted r can ,on xC .J X //Œ1. The flat connection r can commutes with the differentialı and acts by derivations of the Gerstenhaber bracket. Therefore, the de Rham complexDR. xC .J X //Œ1/ WD X ˝ xC .J X //Œ1/ equipped with the differential r can C ı andthe Lie bracket induced by the Gerstenhaber bracket is a sheaf of DGLA on X givingrise to the DGLA g.J X / of global sections.The sheaf of abelian Lie algebras J X =O X acts by derivations of degree 1 on thegraded Lie algebra xC .J X /Œ1 via the adjoint action. Moreover, this action commuteswith the Hochschild differential. Therefore, the (abelian) graded Lie algebra X ˝J X =O X acts by derivations on the graded Lie algebra X ˝ xC .J X //Œ1. We denotethe action of the form ! 2 X ˝ J X=O X by ! . Consider now the subsheaf of closedforms . X ˝J X=O X / cl which is by definition the kernel of r can . .X k ˝J X=O X / cl actsby derivations of degree k 1 and this action commutes with the differential r can C ı.Therefore, for ! 2 .XI . 2 ˝ J X =O X / cl / one can define the !-twist g.J X / ! asthe DGLA with the same underlying graded Lie algebra structure as g.J X / and thedifferential given by r can Cı C ! . The isomorphism class of this DGLA depends onlyon the cohomology class of ! in H 2 ..XI X ˝ J X=O X /; r can /.More precisely, for ˇ 2 .XI X 1 ˝ J X=O X / the DGLA g DR .J X / ! andg DR .J X / !Cr canˇ are canonically isomorphic with the isomorphism depending onlyon the equivalence class ˇ C Im.r can /.As we remarked before a twisted form S of O X is determined up to equivalence by itsclass in H 2 .XI O /. The composition O ! O =C log ! O=C j 1 ! DR.J=O/ inducesthe map H 2 .XI O / ! H 2 .XI DR.J=O// Š H 2 ..XI X ˝J X=O X /; r can /.We denote by ŒS 2 H 2 ..XI X ˝ J X=O X /; r can / the image of the class of S. Bythe remarks above we have the well-defined up to a canonical isomorphism DGLAg DR .J X / ŒS .The rest of this paper is organized as follows. In Section 2 we review some preliminaryfacts. In Section 3 we review the construction of Deligne 2-groupoid, its relationwith the deformation theory and its cosimplicial analogues. In Section 4 we review thenotion of algebroid stacks. Next we define matrix algebras associated with a descentdatum in Section 5. In Section 6 we define the deformations of algebroid stacks andrelate them to the cosimplicial DGLA of Hochschild cochains on matrix algebras. InSection 7 we establish quasiisomorphism of the DGLA controlling the deformationsof twisted forms of O X with a simpler cosimplicial DGLA. Finally, the proof the mainresult of this paper, Theorem 1, is given in Section 8