process

Deformations of gerbes on smooth manifolds 391Let L denote the image of the canonical trivialization under the composition 0 .F /W A B .1/C Š 0 C .1/ F ! 0 C .1/ ŠA B .2/C :Thus, L is a B .1/ ˝R B .2/op -module such that the line bundle L ˝R C is trivial.Therefore, L admits a non-vanishing global section. Moreover, there is an isomorphismf W B .1/01 ˝.B .1/ / 1 .pr 1 1 / L ! .pr 1 0 / 1L˝.B .2/ / 1 B .2/01 of .B.1/ / 1 0 ˝R ..B .2/ / 1 1 /op -0modules.A choice of a non-vanishing global section of L gives rise to isomorphisms B .1/B .1/01 ˝.B .1/ / 1 1The composition.pr 1 1 / L and B .2/01 Š .pr1 0 / L ˝.B .2/ / 1 0B .2/01 .B .1/01 Š B.1/ 01 ˝.B .1/ / 1 .pr 1 1 / L ! f .pr 1 0 / L1˝.B .2/ / 1 B .2/01 Š B.2/ 01001 Šdefines a 1-morphism of deformations of .U; A/ such that the induced 1-morphismC .1/ ! C .2/ is isomorphic to F . This shows that the functor Def 0 .U; A/.R/ !Def.C/ induces essentially surjective functors on groupoids of morphisms. By similararguments left to the reader one shows that these are fully faithful.This completes the proof of Theorem 1.References[1] Revêtements étales et groupe fondamental (SGA 1). Documents Mathématiques (Paris), 3.Société Mathématique de France, Paris, 2003, Séminaire de géométrie algébrique du BoisMarie 1960–61, directed by A. Grothendieck, with two papers by M. Raynaud, updatedand annotated reprint of the 1971 original.[2] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, and D. Sternheimer, Deformation theoryand quantization. I. Deformations of symplectic structures, Ann. Physics 111 (1978),61–110.[3] L. Breen, On the classification of 2-gerbes and 2-stacks, Astérisque 225 (1994).[4] P. Bressler, A. Gorokhovsky, R. Nest, and B. Tsygan, Deformations of Azumaya algebras,in Actas del XVI Coloquio Latinoamericano de Algebra (Colonia del Sacramento, Uruguay,Agosto 2005), Biblioteca de la Revista Matemática Iberoamericana, 2008, 131–152.[5] P. Bressler, A. Gorokhovsky, R.Nest, and B. Tsygan, Deformation quantization of gerbes,Adv. Math. 214 (2007), 230–266.[6] J.-L. Brylinski, Loop spaces, characteristic classes and geometric quantization, Progr.Math. 107, Birkhäuser, Boston, MA, 1993.[7] A. D’Agnolo and P. Polesello, Stacks of twisted modules and integral transforms, in Geometricaspects of Dwork theory, Vol. I, Walter de Gruyter, Berlin 2004, 463–507.[8] J. Dixmier and A. Douady, Champs continus d’espaces hilbertiens et de C -algèbres, Bull.Soc. Math. France 91 (1963), 227–284.[9] M. Gerstenhaber, On the deformation of rings and algebras, Ann. of Math. (2), 79 (1964),59–103.