process

Duality for topological abelian group stacks and T -duality 301Proof. We consider the spectral sequence .F r ;d r / introduced in 4.5.14. It convergesto the graded sheaves associated to a certain filtrations of the cohomology sheaves ofthe total complex of the double complex Hom ShAb S .U ;I / defined in 4.5.7. In degree2; 3; 4 these cohomology sheaves are sheaves of Z=pZ-modules carrying actions ofZ mult with weights determined in Lemma 4.59.The left lower corner of the second page of the spectral sequence was evaluated inLemma 4.60. Note that Ext i Sh Ab S .G; Z/ is a sheaf of Z=pZ-modules with an action ofZ mult of weight 1 for all i 0. The term F 2;12Š Ext 2 Sh Ab S .G; Z/ survives to the limitof the spectral sequence and is a submodule of a sheaf of Z=pZ-modules of weight 2.On the other hand it has weight 1.A Z=pZ-module V with an action of Z mult which has weights 1 and 2 at the sametime must be trivial 11 . In fact, for every q 2 Z mult we get the identity q 2 q D.q 1/q D 0 in End Z .V /, and this implies that q 1.mod p/ for all q 62 pZ. Fromthis follows p D 2, and this case was excluded.Similarly, the sheaf Ext 1 Sh Ab S ..ƒ2 Z G/] ; Z/ is a sheaf of Z=pZ-modules of weight 2(see 3.5.5), and Ext 3 Sh Ab S .G; Z/ is a sheaf of Z=pZ-modules of weight 1. Since p isodd, this implies that the differential d 1;32W Ext 1 Sh Ab S ..ƒ2 Z G/] ; Z/ ! Ext 3 Sh Ab S .G; Z/is trivial. Hence, Ext 3 Sh Ab S .G; Z/ survives to the limit. It is a subsheaf of a sheaf ofZ=pZ-modules which is a weight 2-3-extension. On the other hand it has weight 1.Substituting the weight 1-condition into equation (37), because then P2 v D v v2 D.1 v/v and P3 v D v v3 D .1 v/.1 C v/v this implies that locally every sectionsatisfies .1 v/ n .1 C v/ j v n s D 0 for suitable n; j 2 N (depending on s and theneighborhood) and for all v 2 Z mult . If p>3we can choose v such that .1 v/,.1 C v/, v are simultaneously units, and in this case the equation implies that locallyevery section is zero.We conclude that Ext 3 Sh Ab S .G; Z/ Š 0.Lemma 4.62. Let G be a compact group which satisfies the two-three condition (seeDefinition 4.6). Assume further that1. G is profinite, or2. G is connected and locally topologically divisible (Definition 4.7).Then the sheaves Ext i Sh Ab S .G; Z/ are torsion-free for i D 2; 3.Proof. We must show that for all primes p and i D 2; 3 the maps of sheavesExt i Sh Ab S .G; Z/ ! p Ext i Sh Ab S .G; Z/are injective. The multiplication by p can be induced by the multiplicationp W G ! G:11 Note that in general a Z=pZ-module can very well have different weights. E.g a module of weight khas also weight pk.