Homology and Graphs of Pro-$\mathcal{C}$ Groups

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

$$\mathrm{Cor}^{\mathcal{F}}_{G}: \bigoplus_{t\in T} H_{n}(G_{t} ,B) \longrightarrow H_{n}(G, B), $$

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.

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 $Z$.