process

Duality for topological abelian group stacks and T -duality 277This page contains the object of our interest, namely by 4.2.11 the sheaves of groupsF p;12Š Ext p Sh Ab S.H; Z/:The other spectral sequence .E r ;d r / is obtained by taking the cohomology in theI -direction first. In view of (22) its first page is given byE p;q1Š Ext q Sh Ab S .Z.H p /; Z/;which can be evaluated easily in many cases.Let us note thatH RHom ShZŒH -mod S .Z; Z/ Š Ext Sh ZŒH -mod S .Z; Z/has the structure of a graded ring with multiplication given by the Yoneda product.4.2.13 We now verify Assumption 2 of Lemma 4.9 for all compact groups.Proposition 4.16. Let H 2 S be a compact group. Then we have Ext i Sh Ab S .H ; R/ Š 0for i D 1; 2.As in Definition 4.14 let U WD U .H /. Let R ! I be an injective resolution.Then we get a double complex Hom ShAb S .U ;I /.We first take the cohomology in the I -, and then in the U -direction. We get aspectral sequence with first termE p;q1Š Ext q Sh Ab S .Z.F H p /; R/:It follows from Corollary 3.28, 2., that E p;q1Š 0 for q 1.We consider the complexC .H; R/W 0 ! Map.H; R/ !!Map.H p1 ; R/ ! Map.H p ; R/ !of topological groups which calculates the continuous group cohomology Hcont .H I R/of H with coefficients in R. Now observe that by the exponential law for A 2 SExt 0 Sh Ab S .Z.F H p /; R/.A/Lemma 3.9ŠHom S .H p A; R/ Š Hom S .A; Map.H p ; R//:Hence the complex .E ;01 ;d 1/.A/ is isomorphic to the complexHom S .A; C .H; R// D C .H; R/.A/:Since H is a compact group we have Hcont i .H I R/ Š 0 for i 1. Of importance forus is a particular continuous chain contraction h p W Map.H p ; R/ ! Map.H p 1 ; R/,p 1, which is given by the following explicit formula. If c 2 Map.H p ; R/ is acocycle, then we can define h p .c/ WD b 2 Map.H p 1 ; R/ by the formulaZb.t 1 ;:::;t p 1 / WD . 1/ p c.t 1 ;:::;t p 1 ;t/dt;H