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284 U. Bunke, T. Schick, M. Spitzweck, and A. ThomLemma 4.24. We have D Š colim A.Proof. It suffices to show that J I is cofinal. Let A D be a finitely generatedsubgroup. Choose a finite subset N b B such that the Q-vectorspace Q N b D Qgenerated by N b contains xA and choose representatives b in D of the elements of N b.They generate a finitely generated group U D such that xU D i.U/ D Z N b.Wenowconsider the group G WD hU; Ai. This group is still finitely generated. Similarly, sinceQ xG D Q xU C Q xA D Q N b we haveQ xG \ ZB D Q N b \ ZB D N b xU xG:Moreover, since Q xG D Q N b D Q. xG \ ZB/, Œ xG W xG \ ZB < 1, i.e. G J . On theother hand, by construction A G.On J we define the grading w W J ! N 0 byw.A/ WD jA tors jCrk xA C Œ xA W xA \ ZB for A 2 J:Lemma 4.25. The category J together with the grading w W J ! N 0 is a directcategory in the sense of Definition 5.1.1 in [Hov99].Proof. We must show that A G implies w.A/ w.G/, and that A ¤ G impliesw.A/ < w.G/. First of all we have A tors G tors and therefore jA tors jjG tors j.Moreover we have xA xG, hence rk xA rk xG. Finally, we claim that the canonicalmapxA=. Na \ ZB/ ! xG=. xG \ ZB/is injective. In fact, we have xA \ . xG \ ZB/ D . xA \ xG/ \ ZB D xA \ ZB: It followsthatj xA W xA \ ZBj jxG W xG \ ZBj:Let now A G and w.A/ D w.G/. We want to see that this implies A D G. Firstnote that the inclusion of finite groups A tors ! G tors is an isomorphism since both groupshave the same number of elements. It remains to see that xA ! xG is an isomorphism.We have rk xA D rk xG. Therefore xA \ ZB D Q xA \ ZB D Q xG \ BZ D xG \ BZ.Now the equality Œ xA W xA \ ZB D Œ xG W xG \ ZB implies that xA D xG.4.3.8 The category C.Ab/ has the projective model structure whose weak equivalencesare the quasi-isomorphisms, and whose fibrations are level-wise surjections.By Lemma 4.25 the category J op with the grading w is an inverse category. OnC.Ab/ J opwe consider the inverse model structure whose weak equivalences are thequasi-isomorphisms, and whose cofibrations are the object-wise ones. The fibrationsare characterized by a matching space condition which we will explain in the following.For j 2 J let J j J be the category with non-identity maps all non-identity mapsin J with codomain j . Furthermore consider the functorM j W C.Ab/ J op restriction ! C.Ab/ .J j / op lim ! C.Ab/