process

Duality for topological abelian group stacks and T -duality 295Lemma 4.50. If G is profinite and H is a discrete group, then H i .G; H / is a torsionsheaf for i 1.Proof. We must show that for each section s 2 Hcont i .GI Map.A; H // and a 2 A thereexists a neighbourhood U of a and a number l 2 Z such that .ls/ jU D 0. Thisadditional locality is important. Note that by the exponential law we haveCcont i .G; Map.A; H // WD Hom S.G i ; Map.A; H // Š Hom S .G i A; H /:Let s 2 Hcont i .GI Map.A; H // and a 2 A. Let s be represented by a cycle Os 2Ccont i .G; Map.A; H //. Note that Os W Gi A ! H is locally constant. The sets ¹Os 1 .h/ jh 2 H º form an open covering of G i A.Since G and therefore G i are compact and A 2 S is compactly generated byassumption on S, the compactly generated topology (this is the topology we use here)on the product G i A coincides with the product topology ([Ste67, Theorem 4.3]).Since G i ¹aº G i A is compact we can choose a finite set h 1 ;:::;h r 2 H suchthat ¹Os 1 .h i / j i D 1;:::;rº covers G i ¹aº. Now there exists an open neighbourhoodU A of a such that G i U S rsD1 Os 1 .h i /.OnG i U the function Os has at mosta finite number of values belonging to the set ¹h 1 ;:::;h r º.Since G is profinite there exists a finite quotient group G ! xG such that Os jG i Uhas a factorization Ns W xG i U ! H . Note that Ns is a cycle in Ccont i . xG;Map.U; H //.Now we use the fact that the higher (i.e. in degree 1) cohomology of a finitegroup with arbitrary coefficients is annihilated by the order of the group. Hence j xGjNs isthe boundary of some Nt 2 Ccont i 1.xG;Map.U; H //. Pre-composing Nt with the projectionG i U ! xG i U we get Ot 2 Ccont i .G; Map.U; H // whose boundary is Os. This showsthat .j xGjs/ jU D 0.4.5.7 Let G be a profinite group. We consider the complex U WD U .G/ as definedin 4.14. Let Z ! I be an injective resolution. Then we get a double complexHom ShAb S .U ;I / as in 4.2.12.Lemma 4.51. For i 1H i Hom ShAb S .U ;I / is a torsion sheaf: (36)Proof. We first take the cohomology in the I -, and then in the U -direction. The firstpage of the resulting spectral sequence is given byE r;q1Š Ext q Sh Ab S .Z.Gr /; Z/:Since the G r are profinite topological spaces, by Lemma 3.29 we get E r;q1Š 0 forq 1. We now consider the case q D 0. ForA 2 S we haveE r;01 .A/ Š Z.A Gr / Š Hom S .G r A; Z/ Š Hom S .G r ; Map.A; Z//for all r 0. The differential of the complex .E ;01 .A/; d 1/ is exactly the differential ofthe complex Ccont i .G; Map.A; Z// considered in 4.5.6. By Lemma 4.50 we conclude thatthe cohomology sheaves E i;02are torsion sheaves. The spectral sequence degenerates atthe second term. We thus have shown that H i Hom ShAb S .U ;I / is a torsion sheaf.