process

300 U. Bunke, T. Schick, M. Spitzweck, and A. ThomProof. The proof is similar to the corresponding argument in the proof of 4.16. Using4.2.11 we getF p;q2Š Ext p Sh Ab S ..ƒq Z G/] ; Z/for q D 0; 1; 2. In particular we have ƒ 0 ZG Š Z and thusFurthermore, since ƒ 1 ZG Š G we haveF p;02Š Ext p Sh Ab S.Z; Z/ Š 0:F 0;1Lemma 3.52Š Hom ShAb S .G; Z/ Š Hom top-Ab .G; Z/ Š 0since Hom top-Ab .G; Z/ Š 0 by compactness of G.We claim thatF 0;22Š Hom ShAb S ..ƒ2 Z G/] ; Z/ Š 0:Note that for A 2 S we haveHom ShAb S ..ƒ2 Z G/] ; Z/.A/ Š Hom PrAb S .ƒ2 Z G; Z/.A/ Š Hom Pr Ab S=A.ƒ 2 Z G jA ; Z jA /:An element 2 Hom PrAb S=A.ƒ 2 Z G jA ; Z jA / induces a family of biadditive (antisymmetric)maps W W G.W / G.W / ! Z.W /for .W ! A/ 2 S=A which is compatible with restriction. Restriction to points givesbiadditive maps G G ! Z. Since G is compact the only such map is the constantmap to zero. Therefore W vanishes for all .W ! A/. This proves the claim. Thesame kind of argument shows that Hom ShAb S ..ƒq Z G/] ; Z/.A/ D 0 for q 2.Let us finally show that F 0;32Š Hom ShAb S ..H 3 U / ] ; Z/ Š 0. By 4.15 we have anexact sequence (see (23) for the notation p D )0 ! K ! ƒ 3 Z G ! H 3 . p D / ! C ! 0;where K and C are defined as the kernel and cokernel presheaf, and the middle mapbecomes an isomorphism after tensoring with Q. This means that 0 Š K ˝ Q and0 Š C ˝ Q. Hence, these K and C are presheaves of torsion groups. Let A 2 S. Thenan element s 2 Hom ShAb S ..H 3 U / ] ; Z/.A/ induces by precomposition an element Qs 2Hom ShAb S ..ƒ3 Z G/] ; Z/.A/ Š 0. Therefore s factors over Ns 2 Hom ShAb S .C ] ; Z/.A/.Since C is torsion we conclude that Ns D 0 by Lemma 4.46. The same argument showsthat Hom ShAb S ..H q U / ] ; Z/ Š 0 for q 3.Lemma 4.61. Assume that p is an odd prime number, p ¤ 3. IfG is a compact groupand a Z=pZ-module, then Ext i Sh Ab S .GI Z/ Š 0 for i D 2; 3.If p D 3, then at least Ext 2 Sh Ab S .G; Z/ Š 0.