Ozone Groups of Calabi-Yau and PI AS-Regular Algebras

Definitions and mathematical statement

Let $k$ be a field and $A$ a unital associative $k$-algebra. Its center is \[ Z(A)=\{z\in A:za=az\text{ for every }a\in A\}. \] Under the interpretation that all algebra automorphisms are allowed, define the ozone group by \[ \operatorname{Oz}(A)=\operatorname{Aut}_{Z(A)}(A) =\{\sigma\in\operatorname{Aut}_{k\text{-alg}}(A):\sigma(z)=z\text{ for every }z\in Z(A)\}. \] This gives an explicit interpretation of the supplied notation $\operatorname{Gal}(A/Z(A))$. Triviality means \[ \operatorname{Oz}(A)=\{\operatorname{id}_A\}. \] The record does not specify whether automorphisms must preserve a grading.

For clarity, one standard convention calls $A$ $d$-Calabi--Yau, for an integer $d\geq 0$, if it is homologically smooth and \[ \operatorname{Ext}^{i}_{A^e}(A,A^e)\cong \begin{cases} 0,&i\ne d,\\ A,&i=d, \end{cases} \] as $A$-bimodules, using the usual identification of the resulting module structures, where $A^e=A\otimes_k A^{\mathrm{op}}$. Homological smoothness means that $A$, as an $A^e$-module, admits a bounded resolution by finitely generated projective modules. This convention is supplied to explain the terminology; the record does not specify whether it intends this ungraded definition or a graded variant, nor whether additional finiteness or domain hypotheses apply.

An algebra is PI if there are an integer $m\geq1$ and a nonzero polynomial $f\in k\langle x_1,\ldots,x_m\rangle$ such that $f(a_1,\ldots,a_m)=0$ for every $(a_1,\ldots,a_m)\in A^m$. In a standard connected graded convention, an Artin--Schelter regular algebra is a locally finite ${\mathbb{N}}$-graded algebra $A$ with $A_0=k$, finite global dimension $n$, finite Gelfand--Kirillov dimension, and the Artin--Schelter Gorenstein condition \[ \operatorname{Ext}^{i}_{A}(k,A)\cong \begin{cases}0,&i\ne n,\\k(\ell_L),&i=n,\end{cases} \qquad \operatorname{Ext}^{i}_{A^{\mathrm{op}}}(k,A)\cong \begin{cases}0,&i\ne n,\\k(\ell_R),&i=n,\end{cases} \] for grading shifts $\ell_L,\ell_R$, with $k=A/A_{\geq1}$ and graded Ext understood. The source record does not identify its exact Artin--Schelter regularity conventions, so these conditions must not be treated as recovered source hypotheses.

Requested classifications. With the intended ground field, algebra classes, automorphism convention, and equivalence relation still to be specified, the record asks:

1. Classify the Calabi--Yau algebras $A$ for which every automorphism fixing $Z(A)$ pointwise is the identity.
2. Classify the PI Artin--Schelter regular algebras $A$ whose ozone groups have maximal rank.

The supplied record neither defines "rank" in the second request nor states its maximal value or comparison class. Thus the second request cannot be converted into a precise numerical condition from the supplied evidence. No identification of rank with group order, group-theoretic rank, or module rank is assumed.

The supplied dated curation labels the combined record open, reports that the maximal-rank PI Artin--Schelter regular case is characterized by skew polynomial rings, and reports that the ozone-group abelianity question has been answered. It describes the broader Calabi--Yau classification with trivial ozone group as unresolved. These are supplied literature assessments, not independently verified mathematical conclusions; their hypotheses are not included in the record.

Short English statement

Classify Calabi--Yau algebras whose only automorphism fixing the center pointwise is the identity, and classify PI Artin--Schelter regular algebras with ozone groups of maximal rank. The ground field, precise algebra conventions, classification equivalence, and meaning of maximal rank are unspecified. Supplied dated curation reports a characterization for the second part while retaining the broader first part as open; this draft does not independently verify that assessment.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $30006065$, source problem code catalogue entry, title "Ozone Groups of Calabi--Yau and PI AS-Regular Algebras." The record gives proposed year 2024 and names no individual proposer.
• Supplied workshop attribution: MFO-RIMS Tandem Workshop: Algebraic Geometry and Noncommutative Projective Varieties (2025), Oberwolfach Reports, pp. 2551--2606, DOI 10.4171/owr/2024/44. The original excerpt includes the page marker 2589. Source and literature link: https://doi.org/10.4171/owr/2024/44. The literature list calls this "Oberwolfach Report 44/2024"; the supplied citation's 2025 designation and its 2024 report identification are retained without reconciliation.
• Supplied literature reference: Chan--Gaddis--Won--Zhang, Ozone groups of AS-regular algebras, https://doi.org/10.1007/s00209-025-03777-9.
• Supplied literature reference: Liu--Wu--Zhu, Ozone groups are abelian, https://arxiv.org/abs/2512.21056.
• Literature assessment date: 2026-08-22. This is supplied dated curation, not independently verified. It reports a skew-polynomial-ring characterization of the maximal-rank PI Artin--Schelter regular case, an answer to the ozone-group abelianity question, and an unresolved broader Calabi--Yau classification with trivial ozone group. The background expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says it was reconstructed and checked against the official report. This verification claim is supplied metadata; no source consultation or independent verification is asserted here.