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Duality for topological abelian group stacks and T -duality 2914.5.1 Let0 ! K ! G ! H ! 0be an exact sequence of profinite groups, where K ! G is the inclusion of a closedsubgroup.Lemma 4.38. The sequence of sheavesis exact.0 ! K ! G ! H ! 0Proof. By [Ser02, Proposition 1] every surjection between profinite groups has a section.Hence we can apply Lemma 3.4.In this result one can in fact drop the assumptions that K and G are profinite. In ourbasic example the group K is the connected component G 0 G of the identity of G.Lemma 4.39. Let0 ! K ! G ! H ! 0be an exact sequence of compact abelian groups such that H is profinite. Then thesequence of sheaves0 ! K ! G ! H ! 0is exact.Proof. We can apply [HM98, Theorem 10.35] which says that the projection G ! Hhas a global section. Thus the sequence of sheaves is exact by 3.4 (even as sequenceof presheaves).4.5.2 In order to show that certain discrete groups are not admissible we use thefollowing lemma.Lemma 4.40. Let0 ! K ! G ! H ! 0be an exact sequence of compact abelian groups. If the discrete abelian group yK isadmissible, then sequence of sheavesis exact.0 ! K ! G ! H ! 0Proof. If U is a compact group, then its Pontrjagin dual yU WD Hom top-Ab .U; T/ is adiscrete group. Pontrjagin duality preserves exact sequences. Therefore we have theexact sequence of discrete groups0 ! yH ! yG ! yK ! 0: