Kourovka Notebook Problem 21.78

Definitions and mathematical statement

Fix a prime $p$. A pro-$p$ group is a topological group isomorphic to an inverse limit of finite groups of $p$-power order, with their discrete topologies. Give $\mathbb F_p$ the discrete topology and the trivial $G$-action. Its continuous cochain groups are \[ C^n(G,\mathbb F_p)=\{\hbox{continuous maps }G^n\longrightarrow\mathbb F_p\}. \] Here $C^0=\mathbb F_p$. With trivial coefficients the differential is \[ \begin{split} (df)(g_1,\ldots,g_{n+1})={}&f(g_2,\ldots,g_{n+1})\\ &+\sum_{i=1}^n(-1)^if(g_1,\ldots,g_ig_{i+1},\ldots,g_{n+1})\\ &+(-1)^{n+1}f(g_1,\ldots,g_n). \end{split} \] The continuous cohomology $H^n(G,\mathbb F_p)$ is $\ker d/\operatorname{im}d$ in degree $n$. Cup product, induced on cochains of degrees $r,s$ by \[ (f\smile h)(g_1,\ldots,g_{r+s}) =f(g_1,\ldots,g_r)h(g_{r+1},\ldots,g_{r+s}), \] makes $H^*(G,\mathbb F_p)=\bigoplus_{n\ge0}H^n(G,\mathbb F_p)$ a graded algebra.

Suppose that $H^n(G,\mathbb F_p)$ is finite for every $n\ge0$, equivalently finite-dimensional over $\mathbb F_p$ in each degree. Must this graded algebra be finitely generated over $\mathbb F_p$? Precisely, must there be finitely many homogeneous classes $u_1,\ldots,u_t$ such that every cohomology class is an $\mathbb F_p$-linear combination of products of these classes and the unit? The finite generating list may depend on $G$ and $p$; the hypothesis gives no common dimension bound across degrees.

Short English statement

Does finiteness of every continuous cohomology group of a pro-$p$ group force finite generation of its whole mod-$p$ cohomology algebra?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2587 (KOU-21.78), Kourovka Notebook - New Problems, Issue 21. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 The Kourovka Notebook, 21st edition, version 47 (30 September 2026), new problems, Problem 21.78. Expanded formulation of the supplied catalogue statement. https://arxiv.org/pdf/1401.0300v47.