process

Duality for topological abelian group stacks and T -duality 307where we write n WD Z. n /.Fork 2 Z we define the open subset V k WD N 1 .k/ AG. The family .V k / k2K forms an open pairwise disjunct covering of AG. Since Gis compact the compactly generated topology of A G is the product topology ([Ste67,Theorem 4.3]). Since G is compact, we can choose a finite sequence k 1 ;:::;k r 2 Zsuch that G ¹aº S riD1 V k. Furthermore, we can find a neighbourhood U A of asuch that G U S riD1 V k.Note that N jGU has at most finitely many values. Therefore there exists a finitequotient G ! F such that N jU G factors through f W U F ! Z.The action of Z on G is compatible with the corresponding action of Z on F ,and we have an action n W Hom S .U F;Z/ ! Hom S .U F;Z/. The elementf 2 Hom S .U F;Z/ still satisfies n 2 f D n f . We now take n WD jF jC1. Thenn D id so that .n 2 1/f D 0, hence f D 0. This implies N jU G D 0.This finishes the proof of the second fact.4.5.15 The Lemma 4.62 verifies Assumption 1. in 4.64 for a large class of profinitegroups.Corollary 4.65. If G is a profinite group which satisfies the two-three condition 4.6,then we have Ext i Sh Ab S .G; Z/ Š 0 for i D 2; 3.Theorem 4.66. A profinite abelian group which satisfies the two-three condition isadmissible.4.6 Compact connected abelian groups4.6.1 Let G be a compact abelian group. We shall use the following fact shown in[HM98, Corollary 8.5].Fact 4.67. G is connected if and only if yG is torsion-free.4.6.2 For a space X 2 S and F 2 Pr Ab S let L H .XI F/denote the Čech cohomologyof X with coefficients in F . It is defined as follows. To each open covering Uone associates the Čech complex LC .UI F/. The open coverings form a left-filteredcategory whose morphisms are refinements. The Čech complex depends functoriallyon the covering, i.e. if V ! U is a refinement, then we have a functorial chain mapLC .UI F/! LC .VI F/. We defineandLC .XI F/WD colim ULC .UI F/L H .XI F/WD colim U H . LC .UI F//Š H .colim ULC .UI F// Š H . LC .XI F //: