Gorenstein Equivalences under One-Sided Finiteness

Definitions and mathematical statement

Write $\mathcal A_G$ for the category denoted by $A_G$ in the supplied record. Its definition, the allowed groups $G$, any coefficient data, and the ambient hypotheses of the source's Theorem C are not reproduced. Consequently, the following is a conditional formulation within that source setting, rather than a statement about arbitrary groups or arbitrary abelian categories.

Assuming the usual abelian-category setting with enough projectives and injectives, let $\operatorname{Proj}(\mathcal A_G)$ and $\operatorname{Inj}(\mathcal A_G)$ denote its projective and injective objects. With the standard meanings of the invariant names, set \[ \operatorname{silp}(\mathcal A_G)=\sup\{\operatorname{id}_{\mathcal A_G}P:P\in\operatorname{Proj}(\mathcal A_G)\},\qquad \operatorname{spli}(\mathcal A_G)=\sup\{\operatorname{pd}_{\mathcal A_G}I:I\in\operatorname{Inj}(\mathcal A_G)\}. \] Here $\operatorname{id}$ and $\operatorname{pd}$ are injective and projective dimension, allowing infinite values. These conventions and the required existence of resolutions must be checked against the missing source context.

Let $D^b(\mathcal A_G)$ be the bounded derived category and let $K^b(\operatorname{Proj}(\mathcal A_G))$ and $K^b(\operatorname{Inj}(\mathcal A_G))$ be the homotopy categories of bounded complexes of projectives and injectives, viewed inside the derived category. The two displayed quotient categories in the source fragment are the Verdier quotients \[ D^b(\mathcal A_G)/K^b(\operatorname{Proj}(\mathcal A_G)),\qquad D^b(\mathcal A_G)/K^b(\operatorname{Inj}(\mathcal A_G)). \] A totally acyclic complex of projectives is an exact complex $P^\bullet$ of projective objects for which $\operatorname{Hom}_{\mathcal A_G}(P^\bullet,Q)$ is exact for every projective $Q$. Dually, a totally acyclic complex of injectives is an exact complex $I^\bullet$ of injective objects for which $\operatorname{Hom}_{\mathcal A_G}(J,I^\bullet)$ is exact for every injective $J$. Denote their homotopy categories by $K_{\mathrm{tac}}(\operatorname{Proj}(\mathcal A_G))$ and $K_{\mathrm{tac}}(\operatorname{Inj}(\mathcal A_G))$.

Gorenstein-projective objects are cycles of totally acyclic complexes of projectives; Gorenstein-injective objects are defined dually. Their stable categories, denoted $\underline{\operatorname{GProj}}(\mathcal A_G)$ and $\underline{\operatorname{GInj}}(\mathcal A_G)$, retain these objects and quotient morphisms by those factoring through projectives and injectives, respectively. The supplied fragment explicitly describes the Gorenstein-projective category as Frobenius with projective-injective objects precisely the $\mathcal A_G$-projectives.

The fragment also describes a complete-cohomology category having all objects of $\mathcal A_G$ as objects and zeroth complete cohomology, computed using injective resolutions, as its morphism groups. Its complete-cohomology construction and the first item of the source's category list are missing.

Question, conditional on the unreproduced source hypotheses. For every instance of $\mathcal A_G$ allowed by Theorem C, does that theorem's equivalence conclusion still hold under either one of the separate assumptions \[ \operatorname{silp}(\mathcal A_G)<\infty \quad\text{or}\quad \operatorname{spli}(\mathcal A_G)<\infty, \] without assuming finiteness of the other invariant? The intended conclusion concerns equivalences among the complete-cohomology categories, the two Verdier quotients above, the two homotopy categories of totally acyclic complexes, and the stable Gorenstein-projective and Gorenstein-injective categories. The supplied curation reports that Theorem C establishes its conclusion when both invariants are finite. The exact full category list, comparison functors, and structure preserved by the equivalences cannot be recovered from the supplied fragment; this is therefore a statement-only draft with unresolved source qualifications.

Short English statement

In the setting of the source's Theorem C, ask whether finiteness of either $\operatorname{silp}(\mathcal A_G)$ alone or $\operatorname{spli}(\mathcal A_G)$ alone suffices for its equivalences between complete-cohomology, derived quotient, totally acyclic complex, and stable Gorenstein categories. The supplied record omits the definition of $\mathcal A_G$, the full hypotheses, and part of the conclusion, so a fully specified theorem statement requires source review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005697$, title Gorenstein Equivalences under One-Sided Finiteness; source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied.
• Supplied citation: Mini-Workshop: Homological aspects of TDLC-groups (2024), Oberwolfach Reports, pp. 3103--3138, DOI 10.4171/owr/2023/55. https://doi.org/10.4171/owr/2023/55. The record separately gives proposed year 2023; the relationship between these year labels has not been verified.
• Supplied literature source, labelled "Oberwolfach report": https://ems.press/content/serial-article-files/48166.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and reports that Theorem C is proved assuming both invariants finite, while its ending question asks about one-sided finiteness. This is supplied dated curation, not independently verified literature assessment.
• The registry labels the statement corrected_verified and reports that it was checked against the official report. Those are supplied verification claims; no independent source consultation or mathematical verification is asserted here.