| Popis: |
In the first part of the chapter it is shown that if \(\varLambda\) is a profinite ring and \(M\) is a profinite \(\varLambda\)-module, then each of the functors \(\mathrm{Tor}^{\varLambda}_{n}(M, -)\) commutes with the direct sum of any sheaf of \(\varLambda\)-modules. In particular, if \(G\) is a pro-\(\mathcal{C}\) group, each of its homology group functors \(H_{n}(G, -)\) commutes with any direct sum \(\bigoplus_{t}B_{t}\) of submodules of a \([\![ \varLambda G]\!]\)-module \(B\) indexed continuously by a profinite space, where \([\![ \varLambda G]\!]\) denotes the complete group algebra and \(\varLambda\) is assumed to be commutative. On the other hand, if \(\mathcal{F}= \{G_{t}\mid t\in T\}\) is a continuously indexed family of closed subgroups of \(G\), there is a corestriction map of profinite abelian groups $$\mathrm{Cor}^{\mathcal{F}}_{G}: \bigoplus_{t\in T} H_{n}(G_{t} ,B) \longrightarrow H_{n}(G, B), $$ for all profinite modules \(B\) over \(G\). Using this map one obtains a Mayer-Vietoris exact sequence associated with the action of a pro-\(\mathcal{C}\) group \(G\) on a \(\mathcal{C}\)-tree. |
