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294 U. Bunke, T. Schick, M. Spitzweck, and A. ThomProof. Let C G be a compact generating set of G. Precomposition with the inclusionC ! G gives an inclusion Hom top-Ab .G; H / ! Map.C; H /. WehaveforA 2 SHom ShAb S .G;H/.A/ Š Hom top-Ab.G; H /.A/ (by Lemma 3.5)Š Hom S .A; Hom top-Ab .G; H // Hom S .A; Map.C; H //D Hom S .A C;H/Š H .A C/Š R C .H /.A/:By Lemma 4.44 the sheaf H is a torsion sheaf. It follows from Lemma 4.47 thatR C .H / is a torsion sheaf. By the calculation above Hom ShAb S .G;H/ is a sub-sheafof the torsion sheaf R C .H / and therefore itself a torsion sheaf.Lemma 4.49. If G is a compact abelian group, thenExt 1 Sh Ab S .G; Z/ Š Hom Sh Ab S .G; T/ Š yG:Moreover, if G is profinite, then yG is a torsion sheaf.Proof. We apply the functor Ext Sh Ab S .G;:::/to0 ! Z ! R ! T ! 0and get the following segment of a long exact sequence!Hom ShAb S .G; R/ ! Hom Sh Ab S .G; T/! Ext 1 Sh Ab S .G; Z/ ! Ext1 Sh Ab S .G; R/ ! :Since G is compact we have 0 D Hom top-Ab .G; R/ Š Hom ShAb S .G; R/ by Lemma 3.5.Furthermore, by Proposition 4.16 we have Ext 1 Sh Ab S .G; R/ D 0. Therefore we getExt 1 Sh Ab S .G; Z/ Š Hom Sh Ab S .G; T/ Š Hom top-Ab.G; T/ Š yG;again by Lemma 3.5. If G is profinite, then (and only then) its dual yG is a discretetorsion group by [HM98, Corollary 8.5]. In this case, by Lemma 4.44 the sheaf yG is atorsion sheaf.4.5.6 Let H be a discrete group. For A 2 S we consider the continuous group cohomologyHcont i .GI Map.A; H //, which is defined as the cohomology of the groupcohomology complex0 ! Map.A; H / ! Hom S .G; Map.A; H // !!Hom S .G i ; Map.A; H // !with the differentials dual to the ones given by 4.2.7. The mapS 3 A 7! Hcont i .GI Map.A; H //defines a presheaf whose sheafification we will denote by H i .G; H /.