process

378 P. Bressler, A. Gorokhovsky, R. Nest, and B. Tsyganimplies that for a morphism f W Œp ! Œq in we have f p D f ] q . Therefore thecollection p defines an element in Stack str .G.A/ ˝k m R / 0 . The considerations in5.2.1 and 5.2.2 imply that this construction extends to an isomorphism of 2-groupoidsDef.U; A/.R/ ! Stack str .G.A/ ˝k m R /: (6.1)Combining (6.1) with the embeddingwe obtain the functorStack str .G.A/ ˝k m R / ! Stack.G.A/ ˝k m R / (6.2)Def.U; A/.R/ ! Stack.G.A/ ˝k m R /: (6.3)The naturality properties of (6.3) with respect to base change imply that (6.3) extendsto morphism of functors on the category of commutative Artin algebras.Combining this with the results of Theorems 5.2 and 3.12 implies the following:Proposition 6.4. The functor (6.3) is an equivalence.Proof. By Theorem 5.2 the DGLA G.A/ ˝k m R satisfies the assumptions of Theorem3.12. The latter says that the inclusion (6.2) is an equivalence. Since (6.1) is anisomorphism, the composition (6.3) is an equivalence as claimed.7 JetsIn this section we use constructions involving the infinite jets to simplify the cosimplicialDGLA governing the deformations of a descent datum.7.1 Infinite jets of a vector bundle. Let M be a smooth manifold, and E a locally-freeO M -module of finite rank.Let i W M M ! M , i D 1; 2, denote the projection on the i th factor. Denoteby M W M ! M M the diagonal embedding and let M W O M M ! O M be theinduced map. Let I M WD ker.M /.LetJ k .E/ WD . 1 / O M M =I kC1M ˝2 1O 1M 2 E ;JM k WD Jk .O M /. It is clear from the above definition that JM k is, in a natural way, asheaf of commutative algebras and J k .E/ is a sheaf of JM k -modules. If moreover E isa sheaf of algebras, J k .E/ will canonically be a sheaf of algebras as well. We regardJ k .E/ as O M -modules via the pull-back map 1 W O M ! . 1 / O M MFor 0 k l the inclusion I lC1M! IkC1 M induces the surjective map Jl .E/ !J k .E/. The sheaves J k .E/, k D 0;1;:::together with the maps just defined form aninverse system. Define J.E/ WD limJ k .E/. Thus, J.E/ carries a natural topology.We denote by p E W J.E/ ! E the canonical projection. In the case when E D O Mwe denote by p the corresponding projection p W J M ! O M .Byj k W E ! J k .E/ we