process

264 U. Bunke, T. Schick, M. Spitzweck, and A. Thomassertion. We getdt DDvXd. k .Qt rk / jU W / Dj D1vX k .d Qt rk / jU W Dj D1D s jU W :vX k d.Qt rk / jU Wj D1vX k s jU WCorollary 3.28. 1. If G is discrete, then we have Ext i Sh Ab S .Z.Rn /; G/ Š 0 for alli 1.2. For every compact W 2 S and n 1 we have Ext i Sh Ab S .Z.W /; Rn / Š 0 for alli 1.Lemma 3.29. If W is a profinite space, then every sheaf F 2 Sh Ab S is R W -acyclic.Consequently, Ext i Sh Ab S .Z.W /; F / Š 0 for i 1.Proof. We first show the following intermediate result which is used in 3.4.3 in orderto finish the proof of Lemma 3.29.Lemma 3.30. If W is a profinite space, then .W I :::/is exact.Proof. A profinite topological space can be written as limit, W Š lim I W n , for an inversesystem of finite spaces .W n / n2I . Let p n W W ! W n denote the projections. First of all,W is compact. Every covering of W admits a finite subcovering. Furthermore, a finitecovering admits a refinement to a covering by pairwise disjoint open subsets of the form¹p 1n.x/º x2W nfor an appropriate n 2 I . This implies the vanishing HLp .W; F / Š 0of the Čech cohomology groups for p 1 and every presheaf F 2 Pr Ab S.Let H q D R q i be the derived functor of the embedding i W Sh Ab S ! Pr Ab S ofsheaves into presheaves. We now consider the Čech cohomology spectral sequence[Tam94, 3.4.4] .E r ;d r / ) R .W;F/ with E p;q2Š HLp .W I H q .F // and use[Tam94, 3.4.3] to the effect that HL0 .W I H q .F // Š 0 for all q 1. Combiningthese two vanishing results we see that the only non-trivial term of the second page ofthe spectral sequence is E 0;02Š HL0 .W I H 0 .F // Š F.W/. Vanishing of R i .W I :::/for i 1 is equivalent to the exactness of .W I :::/.3.4.3 We now prove Lemma 3.29. Let i 1. We use that R i R W .F / is the sheafificationof the presheaf S 3 A 7! R i .A W I F/ 2 Ab. For every sheaf F 2 Sh Ab Swe have by some intermediate steps in (16)j D1.A W I F/Š .W I R A .F //:Let us choose an injective resolution F ! I . Using Lemma 3.30 for the secondisomorphism we getH i .AI R W .I // s7!QsŠ H i .A W I I / Qs7!NsŠ .W I H i R A .I //: (18)