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382 P. Bressler, A. Gorokhovsky, R. Nest, and B. TsyganDGLA DR.C .Mat.J.A// p / loc Œ1/ with the underlying sheaf of graded Lie algebras N p U ˝ C .Mat.J.A// p / loc Œ1 and the differential r can C ı.For W Œn ! letG DR .J.A// D .N .n/ UI .0n/ DR.C .Mat.J.A// .0/ / loc Œ1//be the DGLA of global sections. The “inclusion of horizontal sections” map inducesthe morphism of DGLAj 1 W G.A/ ! G DR .J.A// :For W Œm ! Œn in , D ı there is a morphism of DGLA W G DR .J.A// ! G DR .J.A// making the diagramG.A/ j 1 G DR .J.A// G.A/ j 1 G DR .J.A// commutative.Let G DR .J.A// n D Q GŒn! DR .J.A// . The cosimplicial DGLA G DR .J.A//is defined by the assignment 3 Œn 7! G DR .J.A// n , 7! .Proposition 7.7. The map j 1 W G.A/ ! G.J.A// extends to a quasiisomorphismof cosimplicial DGLA.j 1 W G.A/ ! G DR .J.A//:The goal of this section is to construct a quasiisomorphism of the latter DGLA withthe simpler DGLA.The canonical flat connection r can on J X induces a flat connection on xC .J X /Œ1,the complex of O-linear continuous normalized Hochschild cochains, still denotedr can which acts by derivations of the Gerstenhaber bracket and commutes with theHochschild differential ı. Therefore we have the sheaf of DGLA DR. xC .J X /Œ1/ withthe underlying graded Lie algebra X ˝ xC .J X /Œ1 and the differential ı Cr can .Recall that the Hochschild differential ı is zero on C 0 .J X / due to commutativityof J X . It follows that the action of the sheaf of abelian Lie algebras J X D C 0 .J X /on xC .J X /Œ1 via the restriction of the adjoint action (by derivations of degree 1)commutes with the Hochschild differential ı. Since the cochains we consider areO X -linear, the subsheaf O X D O X j 1 .1/ J X acts trivially. Hence the action ofJ X descends to an action of the quotient J X =O X . This action induces the action ofthe abelian graded Lie algebra X ˝ J X=O X by derivations on the graded Lie algebra X ˝ xC .J X /Œ1. Further, the subsheaf . X ˝J X=O X / cl WD ker.r can / acts by derivationwhich commute with the differential ı Cr can .For! 2 .XI .X 2 ˝ J X=O X / cl /