Finite Generation of Group-Ring Cohomology

Definitions and mathematical statement

Use the finite-resolution convention: a group $H$ is of type $\mathrm{FP}$ over ${\mathbb{Z}}$ if its trivial module ${\mathbb{Z}}$ has a finite-length resolution by finitely generated projective ${\mathbb{Z}} H$-modules. A group $\Gamma$ is virtually of this type if it has such a subgroup of finite index. This convention distinguishes $\mathrm{FP}$ from $\mathrm{FP}_\infty$, which permits an infinite resolution with finitely generated terms.

Let ${\mathbb{Z}}\Gamma$ have its left regular $\Gamma$-action and set \[ H^i(\Gamma;{\mathbb{Z}}\Gamma)=\operatorname{Ext}^i_{{\mathbb{Z}}\Gamma}({\mathbb{Z}},{\mathbb{Z}}\Gamma). \] Right multiplication on the coefficient group ring commutes with the left action, giving each cohomology group a right ${\mathbb{Z}}\Gamma$-module structure. Equivalently one can use the left action $\gamma\cdot v=v\gamma^{-1}$. Finite generation is over the group ring, not merely over ${\mathbb{Z}}$.

The question asks whether, for every virtually type-$\mathrm{FP}$ group $\Gamma$, its group-ring cohomology is finitely generated as a $\Gamma$-module. Reading $H^*$ as the total graded module gives the assertion \[ \bigoplus_{i\geq0}H^i(\Gamma;{\mathbb{Z}}\Gamma) =\sum_{k=1}^{N}c_k\,{\mathbb{Z}}\Gamma \quad\text{for some finite set of homogeneous classes }c_1,\ldots,c_N. \] A degreewise reading asks instead for each $H^i(\Gamma;{\mathbb{Z}}\Gamma)$ to be finitely generated. The extract does not explicitly distinguish these readings; they agree when cohomology is concentrated in finitely many degrees. If the intended hypothesis uses $\mathrm{FP}_\infty$ rather than the finite-resolution convention, that change must also be specified. The supplied Coxeter-group result is a special-family qualification, not a replacement of the general virtual-$\mathrm{FP}$ question.

Short English statement

Does every virtually type-$\mathrm{FP}$ group have finitely generated group-ring cohomology as a module over its group ring? The relevant module action, finite-resolution convention and total-versus-degreewise reading are made explicit.

Sources

• Supplied report: Topologie (2007), Oberwolfach Reports, DOI 10.4171/owr/2006/43, pp. 2579--2630; proposed year 2006. The supplied literature reference is labelled "Davis, cohomology with group-ring coefficients": https://doi.org/10.4171/owr/2006/43.
• Local UnsolvedMath registry, permanent ID 30000590, catalogue entry; created 25 August 2026, updated 16 September 2026. Its corrected_verified note describes balancing mathematical notation. The supplied 21 August 2026 literature triage labels the general virtual-$\mathrm{FP}$ question open and reports a Coxeter-group finite-generation result that does not settle this generality. This is a dated assessment requiring independent verification; no further external reference is supplied.