process

Deformations of gerbes on smooth manifolds 385corresponding Čech cohomology is trivial. Therefore, there exists 2 xC 1 .UI J 0 /such that c D L @. Then, 2 Isom 0 .A 01 ˝ J N1 U; J.A 01 //.Suppose that ; 0 2 Isom 0 .A 01 ˝ J N1 U; J.A 01 //. By Corollary 7.4 D 0for some uniquely defined 2 .N 2 UI J 0 /. It is easy to see that 2 xZ 1 .UI J 0 /.We assume from now on that we have chosen 2 Isom 0 .A 01 ˝ J N1 U; J.A 01 //,r2C .A 01 /. Such a choice defines p ij 2 Isom 0.A ij ˝ J Np U; J.A ij // for every pand 0 i;j p by p ij D .prp ij / . This collection of p ijinduces for every p algebraisomorphism p W Mat.A/ p ˝ J Np U ! Mat.J.A// p . The following compatibilityholds for these isomorphisms. Let f W Œp ! Œq be a morphism in . Then thefollowing diagram commutes:f .Mat.A/ p ˝ J Np U/f f ] Mat.A ˝ J Nq U/ q(7.4)f . p /f ] . p /f Mat.J.A// p f f ] Mat.J.A// q .Similarly define the connections r p ijD .pr p ij / r. For p D 0;1;::: set r p D˚pi;jD0 r ij ; the connections r p on Mat.A/ p satisfyf r p D .Ad f /.f ] r q /: (7.5)Note that F.;r/ 2 .N 1 UI 1 N 1 U ˝ J N 1 U/ is a cocycle of degree 1 in LC .UI 1 X ˝J X /. Vanishing of the corresponding Čech cohomology implies that there exists F 0 2.N 0 UI 1 N 0 U ˝ J N 0 U/ such that.d 1 / F 0 .d 0 / F 0 D F.;r/: (7.6)For p D 0;1;:::, 0 i p, let F piiD .pr p i / F 0 ; put F pij D 0 for i ¤ j .Let F p 2 .N p UI 1 N p U ˝ Mat.A/p / ˝ J Np U denote the diagonal matrix withcomponents F pij.Forf W Œp ! Œq we havef F p D f ] F q : (7.7)Then, we obtain the following equality of connections on Mat.A/ p ˝ J Np U:. p / 1 ır can ı p Dr p ˝ Id C Id ˝r can C ad F p : (7.8)The matrices F p also have the following property. Let r can F p be the diagonalmatrix with the entries .r can F p / ii Dr can F pii . Denote by rcan F p the image of r can F punder the natural map .N p UI 2 N p U ˝ Mat.A/p ˝ J Np U/ ! .N p UI 2 N p U ˝Mat.A/ p ˝ .J Np U=O Np U//. Recall the canonical map p W N p U ! X. Then, wehave the following: