Generic Anisotropy of Gorenstein Graded Algebras

Definitions and mathematical statement

Let $k$ be a field. Under the usual commutative convention, a standard graded $k$-algebra is a finitely generated commutative graded algebra \[ A=\bigoplus_{i\geq 0}A_i,\qquad A_0=k,\qquad A=k[A_1]. \] Such an algebra is Gorenstein if, for every prime ideal $\mathfrak p\subseteq A$, the local ring $A_{\mathfrak p}$ has finite injective dimension as a module over itself.

Question 8 (supplied target). Identify classes of Gorenstein standard graded algebras that have the generic anisotropy property.

Unresolved formulation. The supplied record does not define the generic anisotropy property. In particular, it does not specify the generic parameters, any scalar extension or Artinian reduction involved, the elements or graded degrees subject to anisotropy, or the required nonvanishing condition. Consequently, the defining condition for the requested classes cannot be stated precisely from this record alone. The field $k$ above is a parameter for the standard terminology; the target does not specify its characteristic or whether the requested property is to hold over a fixed field, over some field, or over every field. No such quantifier is imposed here. The request is to identify classes; it does not explicitly demand an exhaustive classification.

Adjacent material retained as context. The extracted original text also contains Question 7: "Is any simplicial sphere generically anisotropic over any field?" This is a separate question with an explicit all-fields scope, which cannot automatically be transferred to Question 8. The generic anisotropy terminology is undefined in that adjacent question as well.

The accompanying discussion concerns Lefschetz properties for pseudomanifolds and simplicial cycles. It defines a simplicial complex $D$ of dimension $d-1$ to be $2$-Cohen--Macaulay over $k$ if its Stanley--Reisner ring $k[D]$ is Cohen--Macaulay and, for every vertex $v$, the deletion \[ D\setminus\{v\}=\{\sigma\in D:v\notin\sigma\} \] has dimension $d-1$ and has a Cohen--Macaulay Stanley--Reisner ring over $k$. Here, for a finite simplicial complex $D$ on vertex set $V$, \[ k[D]=k[x_v:v\in V]\big/\left(\prod_{v\in F}x_v:F\subseteq V,\ F\notin D\right). \] These adjacent hypotheses are contextual and are not stated as restrictions on the algebras in Question 8. The extracted discussion refers to Lefschetz-type results in an unidentified reference labelled $[2]$; its bibliographic identity is not supplied.

Short English statement

The problem asks for classes of Gorenstein standard graded algebras with generic anisotropy. The supplied record omits the definition of generic anisotropy and the intended field assumptions, so a fully precise mathematical formulation requires source clarification. The adjacent question about all simplicial spheres over every field is recorded separately.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $30005713$, title Generic Anisotropy of Gorenstein Graded Algebras, exact source code catalogue entry, Question 8. The supplied collection is Oberwolfach Reports Open Problems. No individual proposer is identified; the proposed year is recorded as 2023.
• Supplied report citation: Geometric, Algebraic and Topological Combinatorics (2024), Oberwolfach Reports, pp. 3249--3318, DOI 10.4171/owr/2023/58. Original report: https://doi.org/10.4171/owr/2023/58. This authoritative source is cited by the record; its contents have not been independently consulted for this draft.
• Supplied literature reference: "Fan, generic anisotropy of strongly edge decomposable spheres," https://arxiv.org/abs/2406.18989.
• Supplied literature assessment, dated exactly 2026-08-22: the record labels the problem partially solved, lists characteristic-2 spheres, cycles, and strongly edge decomposable spheres as known classes, and reports that no classification of Gorenstein standard graded algebras with generic anisotropy is known. These are supplied dated curation claims, not independently verified mathematical or literature conclusions. The record explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that it was reconstructed and checked against the official Oberwolfach report. This is supplied verification metadata, not verification performed for this draft.