Duality Invariance of Finite Type Tensor Categories

Definitions and mathematical statement

The supplied question asks whether the finite-type property of finite tensor categories is preserved under duality. The record does not specify the ground field, the definition of finite type, or the intended duality. The following is a conventional formulation requiring confirmation against the cited sources.

Fix an algebraically closed field $k$. A finite tensor category $\mathcal C$ over $k$ is a rigid $k$-linear abelian monoidal category whose underlying abelian category is equivalent to the category of finite-dimensional modules over a finite-dimensional $k$-algebra, whose tensor product is $k$-bilinear and exact in each variable, and whose tensor unit $\mathbf 1$ is simple. Set \[ H^\bullet(\mathcal C,V)=\bigoplus_{n\geq 0}\operatorname{Ext}^n_{\mathcal C}(\mathbf 1,V), \qquad H^\bullet(\mathcal C)=H^\bullet(\mathcal C,\mathbf 1). \] The latter is an algebra under Yoneda composition, and the former is a graded module over it by Yoneda composition. In the cohomological interpretation used here, $\mathcal C$ is of finite type if $H^\bullet(\mathcal C)$ is finitely generated as a $k$-algebra and $H^\bullet(\mathcal C,V)$ is finitely generated over $H^\bullet(\mathcal C)$ for every object $V\in\mathcal C$.

A finite left $\mathcal C$-module category $\mathcal M$ is a finite $k$-linear abelian category equipped with a compatible action of $\mathcal C$. It is exact if $P\otimes M$ is projective in $\mathcal M$ for every projective object $P\in\mathcal C$ and every $M\in\mathcal M$. It is indecomposable if it is not a direct sum of two nonzero $\mathcal C$-module categories. For a nonzero exact indecomposable $\mathcal M$, write \[ \mathcal C_{\mathcal M}^{*}=\operatorname{Fun}_{\mathcal C}^{\mathrm{rex}}(\mathcal M,\mathcal M) \] for the dual tensor category of right exact $\mathcal C$-module endofunctors, with tensor product given by composition and unit the identity functor.

Candidate precise question under these conventions. For every algebraically closed field $k$, every finite tensor category $\mathcal C$ over $k$, and every nonzero exact indecomposable finite left $\mathcal C$-module category $\mathcal M$, does \[ \mathcal C\text{ is of finite type} \quad\Longrightarrow\quad \mathcal C_{\mathcal M}^{*}\text{ is of finite type} \] hold? Here the conclusion includes finite generation with coefficients in every object of $\mathcal C_{\mathcal M}^{*}$.

The universal field quantifier and the module-category interpretation of duality are explicit reconstruction choices, not hypotheses established by the supplied statement. The intended characteristic restrictions, duality convention, and precise finite-generation requirements remain to be checked.

Short English statement

The question asks whether finite generation of tensor-category cohomology, including cohomology with all object coefficients, survives passage to a dual tensor category. The displayed formulation interprets duality through exact indecomposable module categories; the supplied record does not specify these conventions. Its supplied status is partially solved, with an assessment that broad special cases are known while the unrestricted conjecture remains open; this assessment is dated curation, not independently verified here.

Sources

• Supplied local registry record, permanent UnsolvedMath ID $30004098$, source problem code catalogue entry, title Duality Invariance of Finite Type Tensor Categories; collection: Oberwolfach Reports Open Problems. No individual proposer is supplied; the proposed year is recorded as 2019.
• Supplied source citation: Mini-Workshop: Cohomology of Hopf Algebras and Tensor Categories (2020), Oberwolfach Reports, pp. 663--693, DOI 10.4171/owr/2019/11. https://doi.org/10.4171/owr/2019/11. The report is described as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference: Cohomology and duality for finite tensor categories. https://doi.org/10.1090/tran/8548.
• Supplied literature reference: Negron--Plavnik preprint. https://arxiv.org/abs/1807.08854.
• Supplied literature reference: Oberwolfach report. https://doi.org/10.4171/owr/2019/11.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved and reports preservation of finite-type cohomology in several broad settings, including suitable exact module categories and important braided examples, while describing the unrestricted finite-tensor-category conjecture as open. This is supplied dated curation derived from the OWR record, not an independently verified assessment; its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and says its mathematical content was preserved while notation, typography, and sentence structure were repaired for publication. This is supplied editorial metadata, not mathematical verification of this reconstructed formulation.