process

288 U. Bunke, T. Schick, M. Spitzweck, and A. ThomIn particular we have ƒ 0 ZR Š Z and thusF p;02Š Ext p Sh Ab S.Z; Z/ Š 0for p 1 (recall that Ext p Sh Ab S .Z;H/ Š 0 for every H 2 Sh Ab S and p 1).Furthermore, since ƒ 1 ZR Š R, by Lemma 3.5 we haveF 0;12Š Hom ShAb S .R; Z/ Š Hom top-Ab.R; Z/ Š 0since Hom top-Ab .R; Z/ Š 0.Here is a picture of the relevant part of the second page.3 Hom ShAb S ..ƒ3 Z R/] ; Z/2 Hom ShAb S ..ƒ2 Z R/] ; Z/ Ext 1 Sh Ab S ..ƒ2 Z R/] ; Z/1 0 Ext 1 Sh Ab S .R; Z/ Ext2 Sh Ab S .R; Z/ Ext3 Sh Ab S .R; Z/0 Z 0 0 0 0 00 1 2 3 4 54.4.4 Let V be an abelian group. Recall the definition of ƒ V from 4.2.9. If V has thestructure of a Q-vector space, then T 1ZV has the structure of a graded Q-vector space,and I T 1ZV is a graded Q-vector subspace. Therefore ƒ1ZV has the structure ofa graded Q-vector space, too.4.4.5 We claim thatfor i 1. Note thatLet A 2 S. An elementF 0;i2Š Hom ShAb S ..ƒi Z R/] ; Z/ Š 0Hom ShAb S ..ƒi Z R/] ; Z/ Š Hom PrAb S .ƒi ZR; Z/: 2 Hom PrAb S .ƒi Z R; Z/.A/ Š Hom Pr Ab S=A.ƒ i Z R jA ; Z jA /induces a homomorphism of groups W W ƒ i ZR.W / ! Z.W / for every .W ! A/ 2S=A. Since ƒ i ZR.W / is a Q-vector space and Z.W / does not contain divisible elementswe see that W D 0. This proves the claim.The claim implies that F 2;12Š Ext 2 Sh Ab S .R; Z/ survives to the limit of the spectralsequence. Because of (33) it must vanish. This proves Lemma 4.31 in the case i D 2.The term F 1;12Š Ext 1 Sh Ab S .R; Z/ also survives to the limit and therefore alsovanishes because of (33). This proves Lemma 4.31 in the case i D 1.