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Duality for topological abelian group stacks and T -duality 2794.3 Discrete groups4.3.1 In this subsection we study admissibility of discrete abelian groups. First weshow the easy fact that a finitely generated discrete abelian group is admissible. Inthe second step we try to generalize this result using the representation of an arbitrarydiscrete abelian group as a colimit of its finitely generated subgroups. The functorExt Sh Ab S .:::;T/ does not commute with colimits because of the presence of higherR lim-terms in the spectral sequence (26), below.And in fact, not every discrete abelian group is admissible.Lemma 4.17. For n 2 N we have Ext i Sh Ab S .Z=nZ; Z/ Š 0 for i 2.Proof. We apply the functor Ext Sh Ab S .:::;Z/ to the exact sequenceand get the long exact sequence0 ! Z ! Z ! Z=nZ ! 0Ext i 1Sh Ab S .Z; Z/ ! Exti Sh Ab S .Z=nZ; Z/ ! Exti Sh Ab S .Z; Z/ ! :Since by Theorem 4.10 Ext i Sh Ab S .Z; Z/ Š 0 for i 1, the assertion follows.Theorem 4.18. A finitely generated abelian group is admissible.Proof. The group Z=nZ is admissible, since Assumption 2 of Lemma 4.9 followsfrom Proposition 4.16, while Assumption 1 follows from Lemma 4.17. The group Zis admissible by Theorem 4.10 since we can write Z Š Z.¹º/ for a point ¹º2 S. Afinitely generated abelian group is a finite product of groups of the form Z and Z=nZfor various n 2 N. A finite product of admissible groups is admissible.4.3.2 We now try to extend this result to general discrete abelian groups using colimits.Let I be a filtered category (see [Tam94, 0.3.2] for definitions). The categorySh Ab S is an abelian Grothendieck category (see [Tam94, Theorem I.3.2.1]). The categoryHom Cat .I; Sh Ab S/ is again abelian and a Grothendieck category (see [Tam94,Proposition 0.1.4.3]). We have the adjoint paircolim W Hom Cat .I; Sh Ab S/ , Sh Ab S W C ‹ ;where to F 2 Sh Ab S there is associated the constant functor C F 2 Hom Cat .I; Sh Ab S/with value F by the functor C ‹ . The functor C ‹ also has a right-adjoint lim:C ‹ W Sh Ab S , Hom Cat .I; Sh Ab S/ W lim:This functor is left-exact and admits a right-derived functor R lim. Finally, for F 2Hom Cat .I; Sh Ab S/ we have the functorI Hom ShAb S .F;:::/W Sh Ab S ! Hom Cat .I op ; Sh Ab S/