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Duality for topological abelian group stacks and T -duality 2834.3.5 The general remarks on colimits above are true for all sites with finite products(because of the use of Lemma 3.8). In particular we can replace S by the site S lc-acyc oflocally acyclic locally compact spaces.Lemma 4.23. Let F 2 .Sh Ab S lc-acyc / I op . If for all acyclic U 2 S lc-acyc the canonicalmap lim F.U/ ! R lim.F .U // is a quasi-isomorphism of complexes of abelian groups,then lim F ! R limF is a quasi-isomorphism.Proof. We choose an injective resolution F ! I in .Sh Ab S lc-acyc / I op . As in the proofof Lemma 4.22 for each U 2 S lc-acyc the complex I .U / is injective. If we assumethat U is acyclic, the map F.U/ ! I .U / is a quasi-isomorphism in .Ab/ I op , andR lim .F .U // Š lim.I .U //. By assumption.limF /.U / Š lim.F .U // ! lim.I .U // Š .limI /.U / Š .R limF /.U /is a quasi-isomorphism for all acyclic U 2 S lc-acyc . An arbitrary object A 2 S lc-acyc canbe covered by acyclic open subsets. Therefore limF ! limI is an quasi-isomorphismlocally on A for each A 2 S lc-acyc . Hence limF ! R limF is an isomorphism inD C .Sh Ab S lc-acyc / .4.3.6 Let D be a discrete group. Then we let I be the category of all finitely generatedsubgroups of D. This category is filtered. Let F W I ! Ab be the “identity” functor.Then we have a natural isomorphismD Š colim F:By Theorem 4.18 we know that a finitely generated group G is admissible, i.e.R q Hom ShAb S .G; T/ Š 0; q D 1; 2:Note that by Lemma 3.3 we have colimF D D. Using the spectral sequence (26) weget for p D 1; 2 thatR p Hom ShAb S .D; T/ Š Rp lim Hom ShAb S .F ; T/: (27)Let g W S lc-acyc ! S be the inclusion. Since T and D belong to S lc-acyc using Lemma 3.38we also haveg R p Hom ShAb S .D; T/ Š Rp Hom ShAb S lc-acyc.D; T/Š R p (28)lim Hom ShAb S lc-acyc.F ; T/:4.3.7 We now must study the R p lim-term. First we make the index category I slightlysmaller. Let xD D ˝Z Q DW D Q be the image of the natural map i W D ! D Q ,d 7! d ˝ 1. We observe that xD generates D Q . We choose a basis B xD of theQ-vector space D Q and let ZB xD be the Z-lattice generated by B. For a subgroupA D let xA WD i.A/ D Q . We consider the partially ordered (by inclusion) setJ WD ¹A D j A finitely generated and Q xA \ ZB xA and Œ xA W xA \ ZB < 1º:Here ŒG W H denotes the index of a subgroup H in a group G. We still let A denotethe “identity” functor AW J ! Ab.