Dyer–Lashof Finite Generation of Tate Cohomology

Definitions and mathematical statement

Let $G$ be a finite group. After specifying coefficients $k$ and their $G$-action so that the intended cup product is defined, write \[ T_G=\widehat H^{*}(G;k)=\bigoplus_{n\in{\mathbb{Z}}}\widehat H^{n}(G;k) \] for the corresponding graded Tate cohomology ring. Suppose also that a coefficient base for its algebra structure and an action of the intended Dyer--Lashof operations have been specified. For a subset $S\subseteq T_G$, let $\langle S\rangle_{\mathrm{alg},\mathrm{DL}}$ denote the smallest subalgebra over that base which contains $S$ and is closed under all the specified operations wherever they are defined.

The supplied question asks whether, for every finite group $G$, there exists a finite subset $S\subseteq T_G$ such that \[ T_G=\langle S\rangle_{\mathrm{alg},\mathrm{DL}}. \] This expresses the intended finite-generation question only conditionally: the supplied record does not specify the coefficients, a prime, the coefficient action, the version and grading conventions of the Dyer--Lashof operations on Tate cohomology, or the precise meaning of "algebra over the Dyer--Lashof algebra." Closure under multiplication and the specified operations is an explicit interpretation requiring confirmation from the source. No additional quantification over coefficient systems or primes is asserted.

Short English statement

The question asks whether the Tate cohomology ring of every finite group can be generated by finitely many classes using algebra operations and the intended Dyer--Lashof operations. The coefficient system and operation conventions must be recovered before this becomes a fully specified mathematical statement. The supplied record labels the problem open; that status is not independently verified here.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001524$, title "Dyer--Lashof Finite Generation of Tate Cohomology," exact source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. No individual proposer is supplied; the proposed year is recorded as 2010.
• Supplied report citation: Cohomology of Finite Groups: Interactions and Applications (2011), Oberwolfach Reports, pp. 1885--1952, DOI 10.4171/owr/2010/32. https://doi.org/10.4171/owr/2010/32.
• Supplied literature-source label: "Oberwolfach Report 32, Cohomology of Finite Groups." https://doi.org/10.4171/owr/2010/32. The literature assessment dated exactly 2026-08-21 is supplied dated curation, not independently verified: it states that the report explicitly poses the question as open and that the checked literature provides computations and structural results but no general finite-generation theorem or counterexample. The record requests independent verification of time-sensitive status and cited claims.