Periodicity of Stable Auslander Algebras

Definitions and mathematical statement

Under the standard commutative local interpretation, a hypersurface ring is a ring $R\cong S/(f)$, where $S$ is a commutative Noetherian regular local ring and $f$ is a nonzero nonunit nonzerodivisor. The word "singularity" ordinarily restricts attention to nonregular $R$. A finitely generated nonzero $R$-module $X$ is maximal Cohen--Macaulay if \[ \operatorname{depth}_R X=\dim R. \] The ring $R$ has finite Cohen--Macaulay type if there are only finitely many isomorphism classes of indecomposable maximal Cohen--Macaulay $R$-modules. Choose one representative $M_i$ of each such class and put $M=\bigoplus_{i=1}^{t}M_i$. With the endomorphism-ring convention, its stable Auslander algebra is \[ A=\underline{\operatorname{End}}_R(M) =\operatorname{End}_R(M)/\mathcal P_R(M,M), \] where $\mathcal P_R(M,M)$ is the ideal of endomorphisms factoring through finitely generated projective $R$-modules.

If $A$ is a finite-dimensional algebra over a field $k$, the usual meaning of bimodule periodicity is the existence of an integer $p\geq 1$ such that \[ \Omega^p_{A^{\mathrm e}}(A)\cong A, \qquad A^{\mathrm e}=A\otimes_k A^{\mathrm{op}}, \] as $A$-bimodules, where $\Omega^p$ denotes the $p$-th syzygy in a minimal projective $A^{\mathrm e}$-resolution of $A$.

Question. Is the stable Auslander algebra of every hypersurface singularity of finite Cohen--Macaulay type periodic? Under the bimodule interpretation above, this asks whether, for every admissible $R$, there exists $p=p(R)\geq 1$ with $\Omega^p_{A^{\mathrm e}}(A)\cong A$; no common period for all $R$ is requested.

Unresolved qualifications. The supplied record does not specify the coefficient field, its characteristic, completeness or Henselian hypotheses, the dimension of the singularity, or any isolated-singularity hypothesis. It also does not explicitly define "periodic" or state the hypotheses ensuring that the stable Auslander algebra is finite-dimensional over a specified field. Consequently, the definitions above give a conventional conditional interpretation, while the precise intended class of rings and the intended notion of periodicity require source review; these conventions must not be treated as additional verified source hypotheses.

Short English statement

The question asks whether every hypersurface singularity with only finitely many indecomposable maximal Cohen--Macaulay modules has a periodic stable Auslander algebra. Under the usual bimodule interpretation, that algebra must recur as a positive syzygy in its own projective bimodule resolution. The ambient ring hypotheses, coefficient field, and intended meaning of periodicity remain unspecified in the supplied record.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002522$, title Periodicity of Stable Auslander Algebras, exact source code catalogue entry, in the Oberwolfach Reports Open Problems collection. The supplied proposed year is 2014; no individual proposer is identified.
• Representation Theory of Quivers and Finite Dimensional Algebras (2014), Oberwolfach Reports, pp. 453--529, DOI 10.4171/owr/2014/08: https://doi.org/10.4171/owr/2014/08. This is the supplied workshop-report provenance, also listed among the literature sources as "Oberwolfach Report 08/2014."
• Supplied literature source: Periodic resolutions and stable Auslander algebras, https://doi.org/10.1016/j.jpaa.2009.09.012.
• Supplied literature source: Deformed mesh algebras of Dynkin type, https://ora.ox.ac.uk/objects/uuid%3Add7d9fb8-3a52-4b00-937e-ae7b89abc173.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem open and reports periodicity for several finite-type and Dynkin families, including simple plane-curve cases cited in the report, but reports no known general proof for all hypersurface singularities of finite Cohen--Macaulay type. This is supplied dated curation, not an independently verified literature assessment. The background expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and supplies a claim of manual semantic, typographic, and LaTeX review. That supplied label does not establish mathematical correctness or resolve the formulation qualifications recorded here.