Classification of Noncommutative Hypersurfaces

Definitions and mathematical statement

Let $A=\bigoplus_{i\geq 0}A_i$ be an Artin--Schelter regular graded algebra, as stipulated by the supplied reconstructed statement, and let \[ Z(A)=\{z\in A:za=az\text{ for every }a\in A\}. \] For a homogeneous element $f\in Z(A)\cap A_d$, put $B=A/(f)$, where $(f)$ is the two-sided ideal generated by $f$. The record does not specify the base field, the precise convention for Artin--Schelter regularity, the dimension of $A$, or whether $d>0$, $f\neq 0$, and multiplication by $f$ is injective. These are unresolved hypotheses; they are not imposed here without qualification.

Under the usual locally finite, connected graded, right noetherian hypotheses over a field $k$, one defines \[ \operatorname{qgr}A=\operatorname{grmod}A/\operatorname{fdim}A, \] where $\operatorname{grmod}A$ is the category of finitely generated graded right $A$-modules with degree-preserving morphisms, and $\operatorname{fdim}A$ is its Serre subcategory of modules finite dimensional over $k$. Here connected means $A_0=k$, and locally finite means $\dim_k A_i<\infty$ for every $i$. The analogous definition applies to $B$. This explains a standard interpretation of the notation, rather than asserting that the supplied record states all these hypotheses. With this interpretation, restriction of scalars along $A\to B$ gives the hypersurface category denoted in the source by \[ \operatorname{qgr}(A/(f))\subseteq\operatorname{qgr}A. \]

Problem. For the intended class of Artin--Schelter regular graded algebras $A$ and homogeneous central elements $f$, classify the noncommutative hypersurfaces $\operatorname{qgr}(A/(f))$, describe their singularities, and describe their derived categories, extending the understanding available in quadratic and finite-over-the-center cases. This is a classification programme, not a conjecture asserting a specified equivalence or invariant. The record leaves the classification equivalence relation, the meaning of singularity or smoothness, and the choice of derived category unspecified. For example, $D^b(\operatorname{qgr}B)$ denotes the bounded derived category of the abelian category $\operatorname{qgr}B$, but the record does not explicitly select this construction.

The supplied excerpt reports extensive work for quadrics, meaning $d=2$: a Koszul duality criterion for smoothness in its reference [1], descriptions of derived categories in some examples in [2], and a full classification for quadrics in noncommutative surfaces, denoted $n=2$, in [3]. The meaning of $n$ and the hypotheses of these results are absent. These are supplied background claims, not assertions verified by this draft.

For $A$ finite over $Z(A)$, the excerpt attributes to its reference [5] a construction involving a variety $Y$, a finite morphism $Y\to\operatorname{Proj}Z(A)$, and a sheaf of maximal orders $\mathcal A$, with reported equivalences \[ \operatorname{coh}(Y,\mathcal A)\simeq\operatorname{qgr}A, \qquad \operatorname{coh}(V,\mathcal A|_V)\simeq\operatorname{qgr}(A/(f)). \] Here $\operatorname{coh}$ denotes coherent right modules over the indicated sheaf of algebras, and $V\subseteq Y$ is the reported vanishing locus associated with $f$. The excerpt describes $\mathcal A$ as Azumaya on a dense locus, with ramification locus $\Delta\subseteq Y$, and describes restrictions away from $\Delta$ as behaving like possibly twisted ordinary hypersurfaces. The hypotheses needed for these statements and equivalences are not supplied. In particular, the excerpt calls the object defined by homogeneous $f$ a function on $Y$; its projective interpretation may require a section of a twist. The excerpt ends mid-sentence, so no further condition on $V$ can be recovered.

Short English statement

The supplied problem asks for a classification of noncommutative projective hypersurfaces obtained by quotienting an Artin--Schelter regular graded algebra by a homogeneous central element, together with descriptions of their singularities and derived categories. It seeks progress beyond studied quadratic and finite-over-the-center settings. Several hypotheses and the intended classification criteria are missing, and the original excerpt is truncated. The supplied open status is dated curation, not an independently verified conclusion.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30006064$, title "Classification of Noncommutative Hypersurfaces," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No proposer is identified. The record gives proposed year 2024.
• Supplied source citation: MFO-RIMS Tandem Workshop: Algebraic Geometry and Noncommutative Projective Varieties (2025), Oberwolfach Reports, pp. 2551--2606, DOI 10.4171/owr/2024/44. Supplied source and literature link, labelled "Oberwolfach Report 44/2024": https://doi.org/10.4171/owr/2024/44. The citation's 2025 workshop label differs from the report's 2024 designation; this discrepancy is unresolved.
• The supplied original excerpt uses references [1], [2], [3], and [5], but does not provide their bibliographic entries or identify the speaker whose paper is described as [5]. Their identities and precise results cannot be reconstructed from this record.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and reports important special-case results for quadrics and finite-over-the-center orders, while describing the general classification and derived-category programme as open. This is supplied dated curation, not independently verified. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels its statement corrected_verified and says it was reconstructed and checked against the official report. These are supplied provenance claims; this draft does not independently verify them or claim to have visited the report.