Rank Bounds for Modular-Representation Grothendieck Groups

Definitions and mathematical statement

Let $k$ be a finite field of characteristic $p$, and let $G$ be a compact $p$-adic analytic group with no elements of order $p$. Put $d=\dim G$. In the usual Iwasawa-algebra convention, the notation $kG$ here means the completed group algebra \[ A=k[[G]]=\varprojlim_U k[G/U], \] where $U$ ranges over open normal subgroups. This convention must be checked; the ordinary group ring would be a different object. The standard canonical dimension of a nonzero finitely generated module is $d-j_A(M)$, where $j_A(M)=\inf\{i:\operatorname{Ext}_A^i(M,A)\ne0\}$.

Let $\mathcal F_t$ be the source's exact category of finitely generated modules of canonical dimension at most $t$, for $0\leq t\leq d$. Inclusion gives \[ \theta_t:K_0(\mathcal F_t)\longrightarrow K_0(\mathcal F_d). \] Here $K_0$ uses short-exact-sequence relations for these module categories, and $\operatorname{rank}(\operatorname{im}\theta_t)=\dim_{\mathbb{Q}}(\operatorname{im}\theta_t\otimes_{\mathbb{Z}}{\mathbb{Q}})$.

The first question asks to improve the reported upper bound by the number of Galois orbits of conjugacy classes of finite-order elements $g\in G$ with $\dim C_G(g)\leq t$. In symbols, if $\mathcal C_t$ denotes those conjugacy classes and $\mathscr G_k$ the intended Galois action, the bound being refined is read as \[ \operatorname{rank}(\operatorname{im}\theta_t)\leq|\mathscr G_k\backslash\mathcal C_t|. \] Since the torsion orders are prime to $p$, the standard finite-field action is generated by powering $g\mapsto g^{|k|}$; its exact convention requires source confirmation. The source does not specify a sharper proposed formula or what improvement is sought.

The second question asks for an analogue when $k$ is replaced by ${\mathbb{Z}}_p$. In the completed convention the algebra becomes ${\mathbb{Z}}_p[[G]]$; its homological dimension convention and canonical-dimension filtration need to be stated afresh, rather than copying $d$ and the finite-field orbit count unchanged. The record gives no integer-coefficient bound.

The supplied assessment checked 2026-08-21 says both questions remain open. Its additional reference is labeled as a mixed $K(\pi,1)$ paper, whose relevance to these module Grothendieck-group rank questions is not explained; it must not be treated as a matched theorem proving or disproving either requested refinement.

Short English statement

Sharpen the stated Galois-orbit upper bound for the images of canonical-dimension filtration maps on Grothendieck groups, and formulate the corresponding result over ${\mathbb{Z}}_p$. The completed-ring convention and the integer-coefficient filtration must be checked.

Sources

• Local UnsolvedMath registry, permanent ID 30000495, catalogue entry, Rank Bounds for Modular-Representation Grothendieck Groups.
• Pro-$p$ Extensions of Global Fields and pro-$p$ Groups (2007), Oberwolfach Reports, pp. 1463--1536, DOI https://doi.org/10.4171/owr/2006/25.
• Supplied reference "Schmidt on the mixed K-pi-1 case": https://arxiv.org/abs/0801.2103. Its specific relevance to the extracted rank-bound questions is not supplied.
• Supplied direct statement-verification claim restores the hypotheses, filtration map, bound, and coefficient extension; supplied literature triage checked 2026-08-21.