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262 U. Bunke, T. Schick, M. Spitzweck, and A. Thom3.4.2 A sheaf G 2 Sh Ab S is called R W -acyclic if R i R W .G/ Š 0 for i 1.Lemma 3.26. If G is a discrete group, then G is R R n-acyclic.Proof. Let G ! I be an injective resolution. Then we have for A 2 S thatR R n.I /.A/ (16)Š I .A R n /:Therefore R i R R n.G/ is the sheafification of the presheafA 7! H i .I .A R n //:Since G is discrete, the sheaf cohomology of G is homotopy invariant, and thereforeH i .I .A R n // Š H i .A R n I G/ Š Š H i .AI G/:To be precise this can be seen as follows. Let .A/ denote the site of open subsetsof A. It comes with a natural map A W .A/ ! S. The sheaf cohomology functor isthe derived functor of the evaluation functor. In order to indicate on which categorythis evaluation functor is defined we temporarily use subscripts. If I 2 Sh Ab S isinjective, then A I 2 Sh Ab.A/ is still flabby (see [BSS, Lemma 2.32]). This impliesthat H Sh Ab S .AI G/ Š H .A/ .A; A .G//.The diagramA id Aid A 0pr A AA R nis a homotopical isomorphism (in the sense of [KS94, 2.7.4]) A ! A R n over A. Wenow apply [KS94, 2.7.7] which says that the natural mapR.pr A / pr A . A .G// ! R.id A/ id A . A .G//is an isomorphism in D C .Sh Ab .A//. But since G is discrete we get pr A . A G/ Š AR n .G/. If we apply the functor R .A/ .A;:::/to this isomorphism and take cohomologywe get the desired isomorphism marked by Š. 4Now, the sheafification of the presheaf S 3 A ! H i .AI G/ 2 Ab is exactly the ithcohomology sheaf of I which vanishes for i 1.Lemma 3.27. The sheaf R n is R W -acyclic for every compact W 2 S.Proof. Let R n ! I be an injective resolution. Then R i R W .R n / is the sheafificationof the presheafS 3 A 7! H i .Hom ShAb S .Z.W jA /; I jA // Š H i .I .A W //:4 It is tempting to apply a Künneth formula to calculate H .A R n ;G/. But since A is not necessarilycompact it is not clear that the Künneth formula holds.