Cohomology Bounds for Irreducible Finite-Group Modules

Definitions and mathematical statement

Let $G$ be a finite group, $k$ a field, and $M\ne0$ a finite-dimensional absolutely irreducible $kG$-module, meaning that $M\otimes_k\overline{k}$ is irreducible over $\overline{k}G$. Write \[ H^j(G,M)=\operatorname{Ext}^j_{kG}(k,M), \] where $k$ has the trivial $G$-action, and take all dimensions over $k$.

On the uniform reading suggested by the notation $C_j$, the source asks whether \[ \forall j\in{\mathbb{Z}}_{>0}\ \exists C_j\geq0\ \forall(G,k,M)\quad \dim_k H^j(G,M)\leq C_j(\dim_k M)^{j-1}, \] with $C_j$ depending only on $j$. In degree $j=1$, the right-hand side is just $C_1$. The original explicitly includes this degree as conjectural and distinguishes $j=2$ as the case said to be proved in the report.

No faithfulness of the action, restriction on the group, or exclusion of trivial modules is stated in the extraction. These omissions must be checked against the source before treating the displayed unrestricted uniform assertion as the intended conjecture; in particular the literal hypotheses permit one-dimensional trivial modules. No such missing hypothesis is silently added. If dependence of $C_j$ on the field or further fixed data was intended, the quantified statement must be adjusted explicitly.

Short English statement

The extracted question seeks bounds for finite-group cohomology by the $(j-1)$-st power of the dimension of an absolutely irreducible module, uniformly in the group and module. The intended constant dependence and any missing faithfulness or group hypotheses require source review.

Sources

• Local UnsolvedMath registry, permanent ID $30000983$, source code catalogue entry, "Cohomology Bounds for Irreducible Finite-Group Modules." Registry curation: Ulam AI / UnsolvedMath Contributors, CC BY 4.0; underlying sources retain their supplied provenance.
• Profinite and Asymptotic Group Theory (2009), Oberwolfach Reports, pp. 1537--1588, DOI https://doi.org/10.4171/owr/2008/28.
• Supplied "higher-cohomology discussion": https://arxiv.org/abs/0711.2817.
• The original says the report proves j=2 and calls all other positive degrees open, explicitly including j=1. The literature triage checked 2026-08-21 repeats the j=2 claim and says higher-degree cases remain unresolved. These source assessments are preserved as dated attributions, not certified mathematical facts under possibly incomplete hypotheses.