process

Deformations of gerbes on smooth manifolds 381Lemma 7.5. 1. Choose , r as above. Then the differenceF.;r/ D 1 ır canL ı .r ˝Id C Id ˝rcan / (7.1)is an element of 2 1 ˝ End JM.L ˝ J M / Š 1 ˝ J M .2. Moreover, F satisfiesr can F.;r/ C D 0: (7.2)Proof. We leave the verification of the first claim to the reader. The flatness of 1 ıı implies the second claim.r canLThe following properties of our construction are immediateLemma 7.6. 1. Let L 1 and L 2 be two line bundles, and f W L 1 ! L 2 an isomorphism.Let r be a connection on L 2 and 2 Isom 0 .L 2 ˝ J M ; J.L 2 //. ThenF.;r/ D F.Ad f./;Ad f.r//:2. Let L be a line bundle on M , r a connection on L and 2 Isom 0 .L ˝ J M ;J.L//. Let f W N ! M be a smooth map. Thenf F.;r/ D F.f ; f r/:3. Let L be a line bundle on M , r a connection on L and 2 Isom 0 .L ˝ J M ;J.L//. Let 2 .MI J M;0 /. ThenF. ; r/ D F.;r/ Cr can :4. Let L 1 , L 2 be two line bundles with connections r 1 and r 2 respectively, andlet i 2 Isom 0 .L i ˝ J M ; J.L i //, i D 1; 2. ThenF. 1 ˝ 2 ; r 1 ˝ Id C Id ˝r 2 / D F. 1 ; r 1 / C F. 2 ; r 2 /:7.3 DGLAs of infinite jets. Suppose that .U; A/ is a descent datum representing atwisted form of O X . Thus, we have the matrix algebra Mat.A/ and the cosimplicialDGLA G.A/ of local C-linear Hochschild cochains.The descent datum .U; A/ gives rise to the descent datum .U; J.A//, J.A/ D.J.A/; J.A 01 /; j 1 .A 012 //, representing a twisted form of J X , hence to the matrixalgebra Mat.J.A// and the corresponding cosimplicial DGLA G.J.A// of local O-linear continuous Hochschild cochains.The canonical flat connection r can on J.A/ induces the flat connection, still denotedr can on Mat.J.A// p for each p; the product on Mat.J.A// p is horizontal withrespect to r can . The flat connection r can induces the flat connection, still denotedr can on C .Mat.J.A// p / loc Œ1 which acts by derivations of the Gerstenhaber bracketand commutes with the Hochschild differential ı. Therefore we have the sheaf of