process

Deformations of gerbes on smooth manifolds 389Proposition 7.19. The following diagram commutes:G DR .J/ ! .U/ .N/ G DR .J/ ! .V/ˆ;r;FG DR .J.A// .N/ G DR .J.A //.ˆ.N/ ;.N/ r; .N/ F(7.20)8 Proof of the main theoremIn this section we prove the main result of this paper. Recall the statement of thetheorem from the introduction:Theorem 1. Suppose that X is a C 1 manifold and S is an algebroid stack on X whichis a twisted form of O X . Then, there is an equivalence of 2-groupoid valued functorsof commutative Artin C-algebrasDef X .S/ Š MC 2 .g DR .J X / ŒS /:Proof. Suppose U isacoverofX such that 0 S.N 0U/ is nonempty. There is adescent datum .U; A/ 2 Desc C .U/ whose image under the functor Desc C .U/ !AlgStack C .X/ is equivalent to S.The proof proceeds as follows.Recall the 2-groupoids Def 0 .U; A/.R/ and Def.U; A/.R/ of deformations of andstar-products on the descent datum .U; A/ defined in Section 6.2. Note that the compositionDesc R .U/ ! Triv R .X/ ! AlgStack R .X/ induces functors Def 0 .U; A/.R/ !Def.S/.R/ and Def.U; A/.R/ ! Def.S/.R/, the second one being the compositionof the first one with the equivalence Def.U; A/.R/ ! Def 0 .U; A/.R/. We are goingto show that for a commutative Artin C-algebra R there are equivalences1. Def.U; A/.R/ Š MC 2 .g DR .J X / ŒS /.R/ and2. the functor Def.U; A/.R/ Š Def.S/.R/ induced by the functor Def.U; A/.R/ !Def.S/.R/ above.Let J.A/ D .J.A 01 /; j 1 .A 012 //. Then, .U; J.A// is a descent datum for atwisted form of J X . Let R be a commutative Artin C-algebra with maximal ideal m R .Then the first statement follows from the equivalencesDef.U; A/.R/ Š Stack.G.A/ ˝ m R / (8.1)Š Stack.G DR .J.A/// ˝ m R / (8.2)Š Stack.G DR .J X / ŒS ˝ m R / (8.3)Š MC 2 .g DR .J X / ŒS ˝ m R /: (8.4)Here the equivalence (8.1) is the subject of Proposition 6.4. The inclusion of horizontalsections is a quasi-isomorphism and the induced map in (8.2) is an equivalenceby Theorem 3.6. In Theorem 7.18 we have constructed a quasi-isomorphism