Homology and graphs of pro-C groups

In the first part of the chapter it is shown that if Λ is a profinite ring and M is a profinite Λ -module, then each of the functors Torn Λ(M,−) commutes with the direct sum of any sheaf of Λ -modules. In particular, if G is a pro- C group, each of its homology group functors Hn(G, −) commutes with any direct sum ⨁tBt of submodules of a [[ΛG]] -module B indexed continuously by a profinite space, where [[ΛG]] denotes the complete group algebra and Λ is assumed to be commutative. On the other hand, if F= { Gt∣ t∈ T} is a continuously indexed family of closed subgroups of G, there is a corestriction map of profinite abelian groups CorF G: ⨁t∈ T Hn(Gt,B) → Hn(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- C group G on a 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- C product H= ∐z ∈ ZHz, where H is a pro- C group, and { Hz∣ z∈ Z} is a family of closed subgroups of H continuously indexed by a profinite space Z