Uniqueness and containment of virtual blowups, the trivial-image case, and effective descriptions: completing the classification of maximal orders in the 3-dimensional Sklyanin algebra

The 3-dimensional Sklyanin algebra $S$ is the generic noncommutative projective surface, and Hipwood (J. Algebra 602, 2022) classified its connected graded maximal orders: every maximal $S$ -order $U$ with nontrivial image $\overline{U}=(U+gS)/gS$ in $S/gS\cong B(E,\mathcal{L},\sigma)$ is of the form $U=S(\mathbf{x})\cap S$ for a virtual blowup $S(\mathbf{x})$ at a $\sigma$ -virtually effective divisor $\mathbf{x}$ of degree at most 2, where $S(\mathbf{x})$ is an algebra constructed inside the localization $S_{(g)}$ . The paper itself records that this classification is incomplete and its central object is not yet well understood: it is unknown whether $S(\mathbf{x})$ is unique for fixed $\mathbf{x}$ , unknown when $S(\mathbf{x})\subseteq S$ holds, the case $\overline{U}=\mathbb{k}$ is excluded from the classification, no general method produces algebra generators for an arbitrary virtual blowup, and the boundary between virtual blowups that do and do not satisfy the Auslander–Gorenstein and Cohen–Macaulay homological properties is undetermined (an explicit example fails both). The problem is to resolve these questions and thereby complete the classification theory of maximal orders birational to the Sklyanin algebra.

Background

A major goal of noncommutative algebraic geometry is to classify noncommutative analogues of projective surfaces: connected graded (cg) noetherian domains of Gelfand–Kirillov dimension three, studied through their noncommutative projective schemes. Noncommutative planes were classified first — the 3-dimensional Sklyanin algebra $S$ is the generic member of this class. It is a graded $\mathbb{k}$ -algebra on three generators of degree one with quadratic relations parametrized by a triple $(E,\mathcal{L},\sigma)$ , where $E$ is a smooth elliptic curve over an algebraically closed field $\mathbb{k}$ , $\mathcal{L}$ is an invertible sheaf of degree 3, and $\sigma$ is a translation automorphism of $E$ of infinite order. The algebra is Artin–Schelter regular (the noncommutative analogue of a polynomial ring: finite Hochschild dimension, polynomial growth, and Gorenstein duality), and it contains a homogeneous normal element $g$ of degree 3 such that $S/gS\cong B(E,\mathcal{L},\sigma)$ , a twisted homogeneous coordinate ring — the ring built from the spaces of sections $H^0(E,\mathcal{L}^n)$ with multiplication twisted by $\sigma$ , which behaves like the homogeneous coordinate ring of $E$ but with $\sigma$ -twisted grading. Throughout one assumes $S$ is not a finite module over its centre.

By Artin's conjectural picture, understanding algebras birational to $S$ (having the same graded quotient division ring) is the generic case of the surface classification. Birational geometry is studied through maximal orders: a cg domain $R$ inside its graded quotient ring $Q_{\mathrm{gr}}(R)$ is a maximal order if it is maximal, under inclusion, among graded orders equivalent to it — the noncommutative analogue of an integrally closed (normal) variety. Rogalski (2011) introduced the blowup subalgebra $S(p)=\mathbb{k}\langle H^0(E,\mathcal{L}(-p))\rangle$ at a point $p\in E$ (here $\mathcal{L}(-p)$ is $\mathcal{L}$ twisted down by the point) and showed it is the only degree-one-generated maximal order in $S$ ; Rogalski–Sierra–Stafford then classified maximal orders inside the 3-Veronese subring $S^{(3)}=\bigoplus_n S_{3n}$ (the Veronese subring is the graded subring on degrees divisible by 3) as blowups at effective divisors, where a divisor on $E$ is a formal integer combination of points and effective means all coefficients are nonnegative.

Hipwood's thesis (2018) and paper (2022) extended this to maximal orders inside $S$ itself. Two new phenomena appear. First, blowups $S(\mathbf{d})$ at effective divisors $\mathbf{d}$ of degree $\le 2$ need not be generated in a single degree and are maximal orders that are Auslander–Gorenstein and Cohen–Macaulay (homological regularity properties: Auslander–Gorenstein means finite injective dimension with the Auslander condition that $\operatorname{Ext}$ classes never occur in degrees below where they are forced; Cohen–Macaulay means every nonzero graded module attains its homological grade bound). Second, the full classification requires virtual blowups at $\sigma$ -virtually effective divisors: a divisor $\mathbf{x}$ (which may have negative coefficients) is virtually effective if $\mathbf{x}+\sigma^{-1}(\mathbf{x})+\cdots+\sigma^{-(n-1)}(\mathbf{x})$ is effective for all $n\gg 0$ . For such $\mathbf{x}$ , a virtual blowup $S(\mathbf{x})$ is constructed only inside the localization $S_{(g)}$ obtained by inverting powers of $g$ : it is a cg algebra that is a maximal order in $Q_{\mathrm{gr}}(S)$ whose image modulo $g$ agrees with $B(E,\mathcal{L}(-\mathbf{x}),\sigma)$ in all large degrees. The main theorem states that every cg maximal $S$ -order $U$ with $\overline{U}\ne\mathbb{k}$ , where $\overline{U}=(U+gS)/gS$ is the image of $U$ in $S/gS$ , equals $S(\mathbf{x})\cap S$ for such a virtual blowup with $0\le\deg\mathbf{x}\le 2$ , and that these maximal orders are automatically strongly noetherian and satisfy the Artin–Zhang $\chi$ conditions.

The paper is explicit that this leaves the classification theory incomplete in four linked ways: the construction of $S(\mathbf{x})$ does not determine it up to equality inside $S_{(g)}$ , so well-definedness of the correspondence $\mathbf{x}\mapsto S(\mathbf{x})$ is open; it is not known for which $\mathbf{x}$ the virtual blowup actually lies in $S$ rather than only in the localization; maximal orders with $\overline{U}=\mathbb{k}$ (trivial image modulo $g$ ) are excluded from the main theorem by an assumption the sources call "annoying yet necessary" — examples exist, and the thesis's Section 6.2 workaround describes such orders only up to semi-graded isomorphism and passage to a Veronese subring, not the orders themselves; and while the paper constructs the first explicit virtual blowup (realizing it also as an endomorphism ring) and shows it is neither Auslander–Gorenstein nor Cohen–Macaulay, there is no general generator computation for virtual blowups — the naive degree-by-degree choice modelled on $S(\mathbf{d})$ provably fails to give a maximal order — and no delineation of which virtual blowups enjoy the Auslander–Gorenstein or Cohen–Macaulay properties.

Problem Statement

Complete the classification theory of connected graded maximal orders in the 3-dimensional Sklyanin algebra $S$ by resolving the four open questions recorded by Hipwood (J. Algebra 602, 2022; PhD thesis, arXiv:1812.04137), all serving the single objective of making the divisor-to-order correspondence $\mathbf{x}\mapsto S(\mathbf{x})$ complete, well-defined, and effective:

(i) Uniqueness and containment of virtual blowups: for a fixed $\sigma$ -virtually effective divisor $\mathbf{x}$ with $\deg\mathbf{x}\le 2$ , is the virtual blowup $S(\mathbf{x})$ unique up to equality inside the localization $S_{(g)}$ , and under what conditions on $\mathbf{x}$ does the containment $S(\mathbf{x})\subseteq S$ hold?

(ii) The excluded trivial-image case: classify, up to graded isomorphism of subalgebras of $S$ , the connected graded maximal $S$ -orders $U$ (and, for each $d\ge 1$ , the maximal orders inside the degree- $d$ Veronese subring $S^{(d)}=\bigoplus_n S_{dn}$ ) with $\overline{U}=(U+gS)/gS=\mathbb{k}$ — the case excluded from the main classification theorem by an assumption the source calls "annoying yet necessary". Examples exist, and the thesis's Section 6.2 describes such orders only up to semi-graded isomorphism and passage to a Veronese subring; the open target is a classification of these orders themselves.

(iii) Effective descriptions: give a general method to compute algebra generators for an arbitrary virtual blowup $S(\mathbf{x})$ , in particular resolving whether, in general, the two-point blowups $S(p+q)$ are generated in degrees 1 and 2.

(iv) Homological boundary: determine exactly which virtual blowups are Auslander–Gorenstein and Cohen–Macaulay, given that $S(\mathbf{d})$ with $\deg\mathbf{d}\le 2$ always are and that an explicit virtual blowup is neither.

The overarching objective is met when the correspondence between virtually effective divisors and maximal orders is well-defined (i), total (ii), and computable (iii), with its homological regularity delineated (iv).

References

Dominic Hipwood, "Maximal orders in the Sklyanin algebra", Journal of Algebra 602 (2022), 555–598. DOI: 10.1016/j.jalgebra.2022.02.015; arXiv:2008.06947; https://doi.org/10.1016/j.jalgebra.2022.02.015
Dominic Hipwood, "Maximal Order in the Sklyanin Algebra", PhD thesis (supervisor J. T. Stafford), University of Manchester, 2018. arXiv:1812.04137; https://arxiv.org/abs/1812.04137
Daniel Rogalski, "Blowup subalgebras of the Sklyanin algebra", Advances in Mathematics 226 (2011), no. 2, 1433–1473. DOI: 10.1016/j.aim.2010.08.009; arXiv:0912.2304; https://doi.org/10.1016/j.aim.2010.08.009
D. Rogalski, S. J. Sierra, J. T. Stafford, "Noncommutative blowups of elliptic algebras", Algebras and Representation Theory 18 (2015), 491–529. DOI: 10.1007/s10468-014-9506-7; arXiv:1308.2216; https://doi.org/10.1007/s10468-014-9506-7
Daniel Rogalski, Susan Sierra, J. Toby Stafford, "Classifying orders in the Sklyanin algebra", Algebra & Number Theory 9 (2015), no. 9, 2055–2119. DOI: 10.2140/ant.2015.9.2055; arXiv:1308.2213; https://doi.org/10.2140/ant.2015.9.2055
D. Rogalski, S. J. Sierra, J. T. Stafford, "Some noncommutative minimal surfaces", Advances in Mathematics 369 (2020), 107151. DOI: 10.1016/j.aim.2020.107151; arXiv:1807.09889; https://doi.org/10.1016/j.aim.2020.107151
Daniel Rogalski, Susan Sierra, J. T. Stafford, "Ring-theoretic blowing down II: Birational transformations", Journal of Noncommutative Geometry 17 (2023), 1045–1097. DOI: 10.4171/JNCG/510; arXiv:2107.01991; https://doi.org/10.4171/JNCG/510