process

Deformations of gerbes on smooth manifolds 383we denote by DR. xC .J X /Œ1/ ! the sheaf of DGLA with the underlying graded Liealgebra X ˝ xC .J X /Œ1 and the differential ı Cr can C ! . Letg DR .J X / ! D .XI DR. xC .J X /Œ1/ ! /;be the corresponding DGLA of global sections.Suppose now that U isacoverofX; let W N U ! X denote the canonical map.For W Œn ! letG DR .J/ ! D .N .n/UI DR. xC .J X /Œ1/ ! /:For W Œm ! and a morphism W Œm ! Œn in such that D ı the map.m/ ! .n/ induces the map W G DR .J/ ! ! G DR.J/ ! :For n D 0;1;:::let G DR .J/ n ! D Q Œn !G DR .J/ ! . The assignment Œn 7! G DR.J/ n !extends to a cosimplicial DGLA G DR .J/ ! . If we need to explicitly indicate the coverwe will also denote this DGLA by G DR .J/ ! .U/.Lemma 7.8. The cosimplicial DGLA G DR .J/ ! is acyclic, i.e. satisfies the condition(3.8).Proof. Consider the cosimplicial vector space V withV n D .N n UI DR. xC .J X /Œ1/ ! /and the cosimplicial structure induced by the simplicial structure of N U. The cohomologyof the complex .V ;@/is the Čech cohomology of U with the coefficients inthe soft sheaf of vector spaces ˝ xC .J X /Œ1 and, therefore, vanishes in the positivedegrees. G DR .J/ ! as a cosimplicial vector space can be identified with yV in thenotations of Lemma 2.1. Hence the result follows from Lemma 2.1.We leave the proof of the following lemma to the reader.Lemma 7.9. The map W g DR .J X / ! ! G DR .J/ 0 ! induces an isomorphism of DGLAg DR .J X / ! Š ker.G DR .J/ 0 ! G DR.J/ 1 ! /where the two maps on the right are .@ 0 0 / and .@ 0 1 / .Two previous lemmas together with Corollary 3.13 imply the following:Proposition 7.10. Let ! 2 .XI . 2 X ˝ J X=O X / cl / and let m be a commutativenilpotent ring. Then the map induces equivalence of groupoids:MC 2 .g DR .J X / ! ˝ m/ Š Stack.G DR .J/ ! ˝ m/: