process

Deformations of gerbes on smooth manifolds 375For D D .D / 2 F s C n , D 2 c and W Œn1 ! leth n .D/ D X I;ee D I ewhere e W Œs ! .0/ is an injective map, I Wf0;:::;kg!Œs is a surjection, and D I eis the I -component of D e in the sense of the decomposition (5.9). Put h n .D/ D.h n .D/ / 2 C n 1 . It is clear that h n .D/ 2 F s C n 1 . Thus the assignment D 7!h n .D/ defines a filtered map h n W C n ! C n 1 for all n 1. Hence, for any s we havethe map Gr s h n W Gr s C n ! Gr s C n 1 .Next, we calculate the effect of the face maps @ n i 1 W Œn 1 ! Œn. First of all,note that, for 1 i n 1, .@ n i 1 / D Id. The effect of .@ n 0 1 / is the map on globalsections induced by.01/ ] W .1n/ C k .Mat.A/ .1/ / loc ! .0n/ C k .Mat.A/ .0/ / locand .@ n n 1 / D .n 1; n/ . Thus, for D D .D / 2 F s C n ,wehave:X..@ n i 1 / h n .D// D .@ n i1 / e D Ị ı@ n i1 / eOn the other hand,8ˆ0,h nC1 ı @ n C @ n 1 ı h n D Id.Proof of Theorem 5.2. Consider for a fixed k the complex of Hochschild degree kcochains C . It is shown in Proposition 5.5 that this complex admits a finite filtrationF C such that GrC is acyclic in positive degrees. Therefore C is also acyclic inpositive degree. As this holds for every k, the condition (3.8) is satisfied.