process

Duality for topological abelian group stacks and T -duality 287and consider the following part of the resulting long exact sequence for i D 2; 3:Ext i 1Sh Ab S .Z; Z/ ! Exti Sh Ab S .T; Z/ ! Exti Sh Ab S .R; Z/ ! Exti Sh Ab S .Z; Z/:The outer terms vanish since RHom ShAb S .Z;F/ Š F for all F 2 Sh Ab S. Thereforeby Lemma 4.31Ext i Sh Ab S .T; Z/ Š Exti Sh Ab S .R; Z/ Š 0;and this is Assumption 1. of 4.9.It remains to prove Lemma 4.31.4.4.1 Proof of Lemma 4.31.We choose an injective resolution Z ! I . The sheafR gives rise to the homological complex U introduced in 4.14. We get the doublecomplex Hom ShAb S .U ;I / as in 4.2.12. As before we discuss the associated twospectral sequences which compute Ext Sh ZŒR-mod S .Z; Z/.4.4.2 We first take the cohomology in the I -, and then in the U -direction. In viewof (24), the first page of the resulting spectral sequence is given byE p;q1Š Ext q Sh Ab S .Z.F Rp /; Z/;where F R denotes the underlying sheaf of sets of R. By Corollary 3.28, 1. we haveE p;q1Š 0 for q 1.We now consider the case q D 0. ForA 2 S we have E p;01.A/ Š .A R p I Z/ Š.AI Z/ for all p, i.e. E p;01Š Z. We can easily calculate the cohomology of thecomplex .E ;01 ;d 1/ which is isomorphic to0 ! Z 0 ! Z id ! Z 0 ! Z id ! Z ! :We getH i .E ;01 ;d 1/ Š´Z; i D 0;0; i 1:The spectral sequence .E r ;d r / thus degenerates at the second term, and´H i Hom ShAb S .U Z; i D 0;;I / Š0; i 1:(33)4.4.3 We now consider the second spectral sequence .Frp;q ;d r / associated to the doublecomplex Hom ShAb S .U ;I / which takes cohomology first in the U and then in theI -direction. Its second page is given byF p;q2Š Ext p Sh Ab S .H q U ; Z/:Since R is torsion-free we can apply 4.2.11 and getF p;q2Š Ext p Sh Ab S ..ƒq Z R/] ; Z/: