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266 U. Bunke, T. Schick, M. Spitzweck, and A. Thom3.4.5 We have now verified that f W S lc ! S satisfies the assumptions of Proposition3.21.Corollary 3.33. Let f W S lc ! S be the inclusion of the site of locally compact spaces.For F 2 Sh Ab S lc and G 2 Sh Ab S we havef RHom ShAb S .f F;G/ Š RHom ShAb S lc.F; f G/:In particular we havef Ext k Sh Ab S .f F;G/ Š Ext k Sh Ab S lc.F; f G/for all k 0.In fact, the first assertion implies the second since f is exact.We need this result in the following special case. If G 2 S is a locally compact group,then by abuse of notation we write G for the sheaves of abelian groups represented byG in both categories Sh Ab S and Sh Ab S lc .We have f G D G. By Lemma 3.17 we also have f G Š G.Corollary 3.34. Let G; H 2 S lc be locally compact abelian groups. We havef Ext k Sh Ab S .G;H/ Š Extk Sh Ab S lc.G;H/for all k 0.3.4.6 In some places we will need a second sub-site of S, the site S loc-acyc of locallyacyclic spaces.Definition 3.35. A space U 2 S is called acyclic, if H i .U I H / Š 0 for all discreteabelian groups H and i 1.By Lemma 3.30 all profinite spaces are acyclic. The space R n is another exampleof an acyclic space. In fact, the homotopy invariance used in the proof of 3.26 showsthat the inclusion 0 ! R n induces an isomorphism H i .R n I H / Š H i .¹0ºI H /, and aone-point space is clearly acyclic.Definition 3.36. A space A 2 S is called locally acyclic if it admits an open coveringby acyclic spaces.In general we do not know if the product of two locally acyclic spaces is againlocally acyclic (the Künneth formula needs a compactness assumption). In order toensure the existence of finite products we consider the combination of the conditionslocally acyclic and locally compact.Note that all finite-dimensional manifolds are locally acyclic and locally compact.An open subset of a locally acyclic locally compact space is again locally acyclic. Welet S lc-acyc S be the full subcategory of locally acyclic locally compact spaces. Thetopology on S lc-acyc is given bycov Slc-acyc .A/ WD cov S .A/:Let g W S lc-acyc ! S be the inclusion. The proofs of Lemma 3.31, Lemma 3.32 andCorollary 3.33 apply verbatim.