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Let $p$ be a prime and let $G$ be a pro-$p$ group. Suppose that all of the (continuous Galois) cohomology groups $H^n(G, \mathbb{F}_p)$ of $G$ with coefficients in the field of $p$ elements are finite. Does it necessarily follow that the cohomology ring $H^*(G, \mathbb{F}_p)$ is finitely generated?

Contributor: P. Kropholler

Progress

The answer is known to be “yes” if $G$ is abelian-by-($p$-adic analytic), as follows from (J. King, Commun. Algebra, 27, no. 10 (1999), 4969–4991).

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