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292 U. Bunke, T. Schick, M. Spitzweck, and A. ThomThe surjective map of discrete sets yG ! yK has of course a section. Therefore byLemma 3.4 the sequence of sheaves of abelian groups0 ! yH ! yG ! yK ! 0is exact. We apply Hom ShAb S .:::;T/ and get the exact sequence0 ! Hom ShAb S . yK; T/ ! Hom ShAb S . yG; T/! Hom ShAb S . yH ; T/ ! Ext 1 Sh Ab S . yK; T/ ! :By Lemma 3.5 we have Hom ShAb S . yG; T/ Š G etc. Therefore this sequence translatesto0 ! K ! G ! H ! Ext 1 Sh Ab S . yK; T/ ! : (35)By our assumption Ext 1 Sh Ab S . yK; T/ D 0 so that G ! H is surjective.4.5.3 The same argument does apply for the site S lc . Evaluating the surjection G ! Hon H we conclude the following fact.Corollary 4.41. If K G is a closed subgroup of a compact abelian group suchthat yK is admissible or admissible over S lc , then the projection G ! G=K has localsections.4.5.4 If G is an abelian group, then let G tors G denote the subgroup of torsionelements. We call G a torsion group, if G tors D G. IfG tors Š 0, then we say that G istorsion-free. If G is a torsion group and H is torsion-free, then Hom Ab .H; G/ Š 0.A presheaf F 2 Pr S S is called a presheaf of torsion groups if F .A/ is a torsiongroup for every A 2 S. The notion of a torsion sheaf is more complicated.Definition 4.42. A sheaf F 2 Sh Ab S is called a torsion sheaf if for each A 2 S andf 2 F .A/ there exists an open covering .U i / i2I of A such that f jUi 2 F.U i / tors .The following lemma provides equivalent characterizations of torsion sheaves.Lemma 4.43 ([Tam94], (9.1)). Consider a sheaf F 2 Sh Ab S. The following assertionsare equivalent.1. F is a torsion sheaf.2. F is the sheafification of a presheaf of torsion groups.3. The canonical morphism colim n2N . n F/! F is an isomorphism, where . n F/ n2Nis the direct system n F WD ker.F nŠ! F/.Note that a subsheaf or a quotient of a torsion sheaf is again a torsion sheaf.Lemma 4.44. If H is a discrete torsion group, then H is a torsion sheaf.