process

Deformations of gerbes on smooth manifolds 379denote the map e 7! 1 ˝ e, and j 1 WD limj k . In the case E D O M we also have thecanonical embedding O M ! J M given by f 7! f j 1 .1/.Letd 1 W O M M ˝ 2 O M2 1 E ! 1 1 1 M ˝1 1OO M MM˝ 12 O 1M 2 Edenote the exterior derivative along the first factor. It satisfiesd 1 .I kC1M ˝2 1O 1M 2 E/ 1 1 1 M ˝1 1OIk M M ˝2 1O 1M 2 Efor each k and, therefore, induces the mapd .k/1W J k .E/ ! 1 M=P ˝O MJ k 1 .E/:The maps d .k/1for different values of k are compatible with the maps J l .E/ ! J k .E/giving rise to the canonical flat connectionr can W J.E/ ! 1 M ˝ J.E/:Here and below we use notation ./ ˝ J.E/ for lim./ ˝OM J k .E/.Since r can is flat we obtain the complex of sheaves DR.J.E// D . M ˝J.E/; rcan /.When E D O M embedding O M ! J M induces embedding of de Rham complexDR.O/ D . M;d/into DR.J/. We denote the quotient by DR.J=O/. All the complexesabove are complexes of soft sheaves. We have the following:Proposition 7.1. The (hyper)cohomology H i .M; DR.J.E/// Š H i ..M I M ˝J.E//; r can / is 0 if i>0. The map j 1 W E ! J.E/ induces the isomorphism between.E/ and H 0 .M; DR.J.E/// Š H 0 ..M I M ˝ J.E//; r can /7.2 Jets of line bundles. Let, as before, M be a smooth manifold, J M be the sheaf ofinfinite jets of smooth functions on M and p W J M ! O M be the canonical projection.Set J M;0 D ker p. Note that J M;0 is a sheaf of O M modules and therefore is soft.Suppose now that L is a line bundle on M . Let Isom 0 .L ˝ J M ; J.L// denote thesheaf of local J M -module isomorphisms L ˝ J M ! J.L/ such that the followingdiagram is commutative:L ˝ J M J.L/Id˝pp L L.It is easy to see that the canonical map J M ! End JM.L˝J M / is an isomorphism.For 2 J M;0 the exponential series exp./ converges. The compositionJ M;0exp! J M ! End JM.L ˝ J M /