Homological finiteness properties of pro-\(p\) modules over metabelian pro-\(p\) groups. (English) Zbl 1107.20042
Let \(G\) be a pro-\(p\) group. A pro-\(p\) \(\mathbb{Z}_p[\![G]\!]\)-module \(B\) is called of homological type \(FP_m\) over \(\mathbb{Z}_p[\![G]\!]\) if \(B\) has a \(\mathbb{Z}_p[\![G]\!]\)-projective resolution where all modules in dimensions less or equal to \(m\) are finitely generated. The author characterizes the modules of homological type \(FP_m\) in the case when \(G\) is a topologically finitely generated pro-\(p\) group that is an extension of \(A\) by \(Q\), with \(A\) and \(Q\) Abelian, and \(B\) is a finitely generated pro-\(p\) \(\mathbb{Z}_p[\![Q]\!]\)-module that is viewed as a pro-\(p\) \(\mathbb{Z}_p[\![G]\!]\)-module via the projection \(G\to Q\).
The characterization is given in terms of the invariant introduced by J. D. King [J. Lond. Math. Soc., II. Ser. 60, No. 1, 83-94 (1999; Zbl 0952.20020)]. The case \(B=\mathbb{Z}_p\) is exactly the classification of the metabelian pro-\(p\) groups of type \(FP_m\) suggested by King and proved by Kochloukova.
The characterization is given in terms of the invariant introduced by J. D. King [J. Lond. Math. Soc., II. Ser. 60, No. 1, 83-94 (1999; Zbl 0952.20020)]. The case \(B=\mathbb{Z}_p\) is exactly the classification of the metabelian pro-\(p\) groups of type \(FP_m\) suggested by King and proved by Kochloukova.
Reviewer: Andrea Lucchini (Brescia)
MSC:
| 20J05 | Homological methods in group theory |
| 20E18 | Limits, profinite groups |
| 20C07 | Group rings of infinite groups and their modules (group-theoretic aspects) |
| 20F16 | Solvable groups, supersolvable groups |
Keywords:
metabelian pro-\(p\) groups; pro-\(p\) modules; homological type \(FP_m\); projective resolutionsCitations:
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