process

312 U. Bunke, T. Schick, M. Spitzweck, and A. Thom4.6.11 We now observe that the spectral sequence .E r ;d r / is functorial in G. To thisend we use the fact that G 7! U .G/ is a covariant functor from groups G 2 S tocomplexes of sheaves on S. IfG 0 ! G 1 is a homomorphism of topological groups inS, then we get an induced map U q .G 0 / ! U q .G 1 /, namely the map Z.G q 0 / ! Z.Gq 1 /.Under the identification made in 4.6.9 the induced mapExt p Sh Ab S lc.Z.G q 1 /; Z/ ! Extp Sh Ab S lc.Z.G q 0/; Z/goes to the mapH p .G q 1 I Z/ ! H p .G q 0 I Z/induced by the pull-backH p .G q 1 I Z/ ! H p .G q 0 I Z/associated to the map of spaces G q 0 ! Gq 1 .4.6.12 The discussion in 4.6.11 and 4.6.10 shows that the Z mult -module structure G.1/induces one on the spectral sequence .E r ;d r /, and we see that E q;p1has weight p(which is the number of factors yG.1/ contributing to this term).We introduce the notation H k WD H k .Hom ShAb S .U ;I // jSlc . Note that H k has afiltration0 D F 1 H k F 0 H k F k H k D H kwhich is preserved by the action of Z mult . The spectral sequence .E r ;d r / converges tothe associated graded sheaf Gr.H k /.Note that Gr p .H k / is a subquotient of E k p;p2. Since a subquotient of a sheaf ofweight p also has weight p (see the our remark after Lemma 4.74) we get the followingconclusion.Corollary 4.75. Gr p .H k / has weight p.4.6.13 We now study some aspects of the second page E q;p2of the spectral sequencein order to show the following lemma.Lemma 4.76. 1. For k 1 we have F 0 H k Š 0.2. For k 2 we have F 1 H k Š 0.Proof. We start with 1. We see that E ;02is the cohomology of the complex E ;01ŠHom ShAb S .U ; Z/. Explicitly, using in the last step the fact that G and therefore G qare connected,E q;01Š Hom ShAb S .U q ; Z/ (24)Š Hom ShAb S .Z.Gq /; Z/ (17)Š R G q.Z/Š Map.G q ; Z/ Š Z (by Lemma 3.25):