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Published in:
2017 | OriginalPaper | Chapter
9. Homology and Graphs of Pro-\(\mathcal{C}\) Groups
Author : Luis Ribes
Published in: Profinite Graphs and Groups
Publisher: Springer International Publishing
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Abstract
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 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.
$$\mathrm{Cor}^{\mathcal{F}}_{G}: \bigoplus_{t\in T} H_{n}(G_{t} ,B) \longrightarrow H_{n}(G, B), $$
When \(G\) is a pro-\(p\) group, this chapter contains a theorem characterizing in terms of the corestriction map when \(G\) is the free pro-\(p\) product of a family of closed subgroups continuously indexed by a profinite space. Using this characterization one proves a Kurosh-type theorem describing the structure of second-countable pro-\(p\) subgroups of a free pro-\(\mathcal{C}\) product \(H = \coprod_{z\in Z} H_{z}\), where \(H\) is a pro-\(\mathcal{C}\) group, and \(\{H_{z}\mid z\in Z\}\) is a family of closed subgroups of \(H\) continuously indexed by a profinite space \(Z\).