Nonvanishing of Derived Categories for Gorenstein Algebras

Definitions and mathematical statement

For a triangulated category $\mathcal T$, write $\mathcal T\neq 0$ if it contains an object not isomorphic to its zero object; equivalently, \[ \exists X\in\operatorname{Ob}(\mathcal T)\quad X\not\cong 0. \] Let $\mathcal G$ denote the class of algebras intended by the source's term "Artin--Schelter Gorenstein algebra." The supplied question is the following nonvanishing assertion: \[ \forall A\in\mathcal G,\qquad D_{\varepsilon}(A)\neq 0, \] or, equivalently, \[ \forall A\in\mathcal G\;\exists X\in\operatorname{Ob}\left(D_{\varepsilon}(A)\right)\quad X\not\cong 0. \] The supplied record describes $D_{\varepsilon}(A)$ as a triangulated category but does not define its objects, morphisms, construction, or the notation $\varepsilon$. It also does not specify the base field or ring, grading conventions, finiteness hypotheses, or the precise definition of Artin--Schelter Gorenstein used in the source. Thus the displayed assertion preserves the source question but remains a formulation schema pending these definitions; no additional hypotheses or conventions are imposed here.

Short English statement

Does every Artin--Schelter Gorenstein algebra $A$, with the conventions intended in the cited source, have an associated category $D_{\varepsilon}(A)$ containing an object not isomorphic to zero? The category and algebra conventions are missing from the supplied record.

Sources

• Supplied local registry record: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30002591$, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is identified; the supplied proposed year is 2014.
• Supplied report citation: Interactions between Algebraic Geometry and Noncommutative Algebra (2015), Oberwolfach Reports, pp. 1365--1402, DOI 10.4171/owr/2014/25. https://doi.org/10.4171/owr/2014/25. This is also the supplied literature source labeled "Oberwolfach Report 25."
• Supplied literature source: Reyes--Rogalski--Zhang, "skew Calabi--Yau categories." https://doi.org/10.1090/tran/6640.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the question open and reports that the authors prove nonvanishing in broad classes, leave nonvanishing for all Artin--Schelter Gorenstein algebras as a conjecture, and that no later general proof was located. This is supplied dated curation derived from the Oberwolfach material, not an independently verified literature assessment. The cited contents and these claims require independent checking.