process

276 U. Bunke, T. Schick, M. Spitzweck, and A. Thom4.2.8 For A 2 S the complex p D .A/ (see (23)) is exactly the standard complex (see4.2.7) for the group homology H .H.A/; Z/ of the group H.A/. The cohomologysheaves H .U / are thus the sheafifications of the cohomology presheavesS 3 A 7! H .H.A/; Z/ 2 Ab:4.2.9 In this paragraph we collect some facts about the homology of abelian groups.An abelian group V is the same thing as a Z-module. We define the graded Z-algebraƒ ZV as the quotient of the tensor algebraT Z V WD M n0V ˝Z ˝Z V„ ƒ‚ …n factorsby the graded ideal I T Z V generated by the elements x ˝ x, x 2 V .4.2.10 Let G be an abelian group. We refer to [Bro82, V.6.4] for the following fact.Fact 4.15. There exists a canonical mapmW ƒ i Z G ! H i.GI Z/:It is an isomorphism for i D 0; 1; 2, and it becomes an isomorphism after tensoring withQ for all i 0. IfG is torsion-free, then it is an isomorphism mW ƒ i Z G ! H i .GI Z/for all i 0.4.2.11 Let U D U .H / (see Definition 4.14) for H 2 Sh Ab S. The cohomologysheaves L coinv.Z// Š H .U / are the sheafifications of the presheaves H . p D /.By the fact 4.15 we have H i . p D / Š ƒ i ZH for i D 0; 1; 2 for the presheaves S 3U 7! ƒ i Z H.U/. In particular, we have H 0 .U / Š Z and H 1 .U / Š H . If H isa torsion-free sheaf, then H i .U / Š .ƒ i Z H/] for all i 0. Finally, for an arbitrarysheaf H 2 Sh Ab S we have.ƒ Z H/] ˝Z Q Š H .U / ˝Z Q:4.2.12 Our application of this relies on the study of the two spectral sequences convergingtoExt Sh Ab S .Lcoinv.Z/; Z/ Š Ext Sh ZŒH -mod S .Z; Z/:We choose an injective resolution Z ! I in Sh Ab S. ThenExt Sh Ab S .Lcoinv.Z/; Z/ Š H .Hom ShAb S .U ;I //:The first spectral sequence denoted by .F r ;d r / is obtained by taking the cohomologyin the U -direction first. Its second page is given byF p;q2Š Ext p Sh Ab S .Lq coinv.Z/; Z/: