process

390 P. Bressler, A. Gorokhovsky, R. Nest, and B. TsyganG.J X / ŒS ! G.J.A//; the induced map (8.3) is an equivalence by another applicationof Theorem 3.6. Finally, the equivalence (8.4) is shown in Proposition 7.10We now prove the second statement. We begin by considering the behavior ofDef 0 .U; A/.R/ under the refinement. Consider a refinement W V ! U . Recallthat by Proposition 7.10 the map induces equivalences MC 2 .g DR .J X / ! ˝ m/ !Stack.G DR .J/ ! .U/ ˝ m/ and MC 2 .g DR .J X / ! ˝ m/ ! Stack.G DR .J/ ! .V/ ˝ m/.It is clear that the diagramMC 2 .g DR .J X / ! ˝ m/ .N/Stack.G DR .J/ ! .U/ ˝ m/ Stack.G DR .J/ ! .V/ ˝ m/commutes, and therefore .N/ W Stack.G DR .J/ ! .U/˝m/ ! Stack.G DR .J/ ! .V/˝m/ is an equivalence. Then Proposition 7.19 together with Theorem 3.6 implies that.N/ W Stack.G DR .J.A/// ˝ m/ ! Stack.G DR .J.A /// ˝ m/ is an equivalence.It follows that the functor W Def.U; A/.R/ ! Def.V; A /.R/ is an equivalence.Note also that the diagramDef.U; A/.R/ Def.V; A /.R/Def 0 .U; A/.R/ Def 0 .V; A /.R/is commutative with the top horizontal and both vertical maps being equivalences.Hence it follows that the bottom horizontal map is an equivalence.Recall now that the embedding Def.U; A/.R/ ! Def 0 .U; A/.R/ is an equivalenceby Proposition 6.3. Therefore it is sufficient to show that the functor Def 0 .U; A/.R/ !Def.S/.R/ is an equivalence. Suppose that C is an R-deformation of S. It follows fromLemma 6.2 that C is an algebroid stack. Therefore, there exists a cover V and an R-descent datum .V; B/ whose image under the functor Desc R .V/ ! AlgStack R .X/ isequivalent to C. Replacing V by a common refinement of U and V if necessary we mayassume that there is a morphism of covers W V ! U. Clearly, .V; B/ is a deformationof .V; A /. Since the functor W Def 0 .U; A/.R/ ! Def 0 .V; A / is an equivalencethere exists a deformation .U; B 0 / such that .U; B 0 / is isomorphic to .V; B/. LetC 0 denote the image of .U; B 0 / under the functor Desc R .V/ ! AlgStack R .X/. Sincethe images of .U; B 0 / and .U; B 0 / in AlgStack R .X/ are equivalent it follows thatC 0 is equivalent to C. This shows that the functor Def 0 .U; A/.R/ ! Def.C/.R/ isessentially surjective.Suppose now that .U; B .i/ /, i D 1; 2, are R-deformations of .U; A/. Let C .i/denote the image of .U; B .i/ / in Def.S/.R/. Suppose that F W C .1/ ! C .2/ is a1-morphism.