process

310 U. Bunke, T. Schick, M. Spitzweck, and A. ThomOf course, this formulation is much too complicated since Z has cohomologicaldimension one. In fact, the spectral sequence decomposes into a collection of shortexact sequences.Lemma 4.72. If G is a compact connected abelian group, then H .GI Z/ Š HL .GI Z/.Proof. The Čech cohomology spectral sequence provides the mapL H .GI Z/ ! H .GI Z/:We now use that fact that G is the projective limit of groups isomorphic to T a , a 2 N.Since the compact abelian group G is connected, by Fact 4.67 we know that yG istorsion-free. Since yG is torsion-free it is the filtered colimit of its finitely generatedsubgroups yF .If yF yG is finitely generated, then yF Š Z a for some a 2 N, thereforeF WD yF Š T a . Pontrjagin duality transforms the filtered colimit yG Š colim y F y F intoa strict limit G Š limF y F .Since T a is a manifold it admits a cofinal system of good open coverings where allmultiple intersections are contractible. Therefore we get the isomorphismL H .T a I Z/ ! H .T a I Z/:Since both cohomology theories commute with limits of compact spaces we getL H .GI Z/ ! H .GI Z/:4.6.7 We now use the calculation of cohomology of the underlying space of a connectedcompact abelian group G. By [HM98, Theorem 8.83] we haveL H .G; Z/ Š ƒ Z y G (41)as a graded Hopf algebra. This result uses the two properties 4.68 and 4.69 of Čechcohomology in an essential way.By Lemma 4.72 we also haveH .GI Z/ Š ƒ Z y G:Note that ƒ Z y G is torsion free. In fact, ƒ Z y G Š colim y F ƒ Z y F , where the colimitis taken over all finitely generated subgroups yF of yG. We therefore get a colimit ofinjections of torsion-free abelian groups, which is itself torsion-free.4.6.8 Recall the notation related to Z mult -actions introduced in 3.5.Lemma 4.73. If V;W 2 Ab with W torsion-free and k; l 2 Z, k 6D l, thenHom Zmult -mod.V .k/; W.l// D 0: