Completing the classification of orders in the Sklyanin algebra: the exceptional case $\overline{U}=k$, Veronese maximality of maximal orders (Question 9.4), and uniqueness of virtual blowups

The classification by Rogalski, Sierra and Stafford of connected graded orders in Veronese rings $T$ of the generic 3-dimensional Sklyanin algebra describes every maximal $T$-order with $\overline{U}\neq k$ as a (virtual) blowup at a virtually effective divisor on the associated elliptic curve, but leaves five explicit residues: the exceptional class with $\overline{U}=k$ (Example 10.8), whether every Veronese $U^{(n)}$ of a maximal order is again a maximal order (Question 9.4), whether the blowup at a virtually effective divisor is unique up to isomorphism (Remarks 6.10(i) and 10.7(2)), whether the technical Assumption 8.2 (existence of a minimal sporadic ideal plus a countability condition) holds beyond the principal Sklyanin families, and the conjecture that blowups generically have no sporadic ideals (Remark 6.10(iii)). Later work of Hipwood resolves Question 9.4 positively under the hypothesis $\overline{U}\neq k$; the remaining core, in particular the $\overline{U}=k$ class, Veronese maximality there, and blowup uniqueness, is open.

Background

A central problem of noncommutative algebraic geometry is the classification of noncommutative projective surfaces, which are studied through connected graded (cg) algebras: $\mathbb{N}$-graded $k$-algebras $A=\bigoplus_{n\geq0}A_n$ over an algebraically closed field $k$ with $A_0=k$ and $\dim_k A_n<\infty$. The generic case is expected to be governed by the 3-dimensional Sklyanin algebra $S=S(a,b,c)$, the cg algebra on three generators with three quadratic relations parametrized by $[a,b,c]\in\mathbb{P}^2$, which for generic parameters deforms $k[x,y,z]$ and whose noncommutative projective plane is the generic one. The algebra $S$ contains a central element $g\in S_3$ with $S/gS\cong B(E,\mathcal{L},\sigma)$, a twisted homogeneous coordinate ring of an elliptic curve $E$ (the ring built from the sections of powers of an invertible sheaf $\mathcal{L}$ with multiplication twisted by an automorphism $\sigma$ of infinite order). One works with the 3rd Veronese $T=S^{(3)}=\bigoplus_n S_{3n}$, regraded so that $g\in T_1$; then $T/gT\cong B(E,\mathcal{M},\tau)$ with $\deg\mathcal{M}=\mu=9$.

For a cg domain $A$ of finite Gelfand-Kirillov dimension (a growth measure), the nonzero homogeneous elements form an Ore set, and inverting them gives the graded quotient ring $Q_{gr}(A)\cong D_{gr}(A)[t,t^{-1};\alpha]$, a skew Laurent polynomial ring over the function skewfield $D_{gr}(A)$. Two orders $U,U'$ with the same quotient ring are equivalent orders if $aUb\subseteq U'$ and $a'U'b'\subseteq U$ for some nonzero $a,b,a',b'$ from the quotient ring. A cg order $U$ is a maximal order if no strictly larger equivalent order lies inside $Q_{gr}(U)$ — the noncommutative analogue of an integrally closed domain, or geometrically of a normal variety — and a maximal $T$-order if no such order lies between $U$ and $T$. The $n$-th Veronese of $U$ is $U^{(n)}=\bigoplus_i U_{ni}$.

Rogalski first classified the maximal orders in $T$ generated in degree one as the blowup subalgebras $T(\mathbb{d})$, constructed from the right ideal associated to an effective divisor $\mathbb{d}$ on $E$ of degree at most $\mu-2$. Rogalski, Sierra and Stafford (RSS) then classified, in the framework of the companion paper on noncommutative blowups of elliptic algebras, all cg maximal $T$-orders $U$ whose image $\overline{U}=(U+gT)/gT$ in $T/gT$ is not just $k$: there is a unique maximal order $F(U)\supseteq U$ equivalent to $U$, and $F(U)$ is a blowup of $T$ at a virtually effective divisor $\mathbb{d}'=\mathbb{d}-\mathbb{y}+\tau^{-1}(\mathbb{y})$ (effective divisors $\mathbb{d},\mathbb{y}$, $0\leq\deg\mathbb{d}'\leq\mu-1$) on $E$. A blowup here is an endomorphism ring $\operatorname{End}_{T(\mathbb{d})}(M)$ of a suitable module whose factor ring is, in large degree, the twisted homogeneous coordinate ring $B(E,\mathcal{M}(-\mathbb{d}'),\tau)$. Strikingly, such $U$ are automatically strongly noetherian (noetherian after any commutative noetherian base extension) and have balanced dualizing complexes. A graded ideal $I$ of $T$ is a sporadic ideal if $\operatorname{GKdim}(T/I)=1$; the theory requires Assumption 8.2, namely that $T$ have a minimal sporadic ideal and that the group of divisors $\operatorname{Div}(T\otimes_k K)$ be countable after base change to an uncountable algebraically closed field $K$ — verified for the quadratic and cubic Sklyanin families, unknown for other families such as the Veroneses of Stephenson's generic two-generator algebras associated to $E$.

The classification has explicit holes, all recorded by the authors. First, the hypothesis $\overline{U}\neq k$ is necessary: for $T=k\langle a_1,\dots,a_r\rangle$, the ring $U=T^g\langle g\rangle$ with $T^g=k\langle ga_1,\dots,ga_r\rangle$ is a maximal order with $\overline{U}=k$, and $\widehat{U}=T$ is not finitely generated over $U$ (Example 10.8), so the divisor-on-$E$ machinery, which reads all information from $\overline{U}$, degenerates for such rings. Second, Question 9.4 asks whether each Veronese $U^{(n)}$ of a cg maximal $T$-order or maximal order is again a maximal $T^{(n)}$-order; RSS prove only that for a noetherian maximal order some Veronese, after a semi-graded re-embedding (an injective homomorphism that multiplies degrees by a fixed factor), lands outside $k+gT$ and is then a maximal order. Third, the blowup at a virtually effective divisor is not unique as a subalgebra of $T$ (different modules can give distinct blowups with the same large-degree factor), and uniqueness up to isomorphism is unknown. Fourth and fifth, the generality of Assumption 8.2 and the conjectured generic absence of sporadic ideals (which would simplify the order pair to $U=F$) are unsettled.

Problem Statement

Complete the classification of connected graded orders in Veronese rings of the generic 3-dimensional Sklyanin algebra, by resolving the residues explicitly recorded by Rogalski-Sierra-Stafford. The objective comprises the five components (A)-(E) below. The components serve one objective — make the classification of maximal orders and maximal $T$-orders birational to $T=S^{(3)}$ complete and its blowup parametrization faithful — and are:

(A) Exceptional case: give a structure theorem for cg maximal $T$-orders (and maximal orders $U\subseteq T$ with $Q_{gr}(U)=Q_{gr}(T)$) whose reduction $\overline{U}=(U+gT)/gT$ equals $k$, the case excluded from the main classification theorem; in particular, decide whether this class consists exactly of rings of the Example 10.8 type $U=T^g\langle g\rangle$ and its analogues, or exhibit and classify further ones.

(B) Veronese maximality (Question 9.4): suppose $U\subseteq T$ is a cg maximal $T$-order or a maximal order, including the case $\overline{U}=k$ and including families $T$ beyond the quadratic Sklyanin Veronese that satisfy the standing Assumption 2.1; is each Veronese ring $U^{(n)}$ a maximal $T^{(n)}$-order? (The sub-case $\overline{U}\neq k$ for the quadratic Sklyanin Veronese is settled by Hipwood and may be cited rather than re-derived; the open part of this component is the case $\overline{U}=k$ together with the families beyond the quadratic Sklyanin Veronese.)

(C) Uniqueness of virtual blowups: for a virtually effective divisor on the elliptic curve, the blowups $F$ whose factors are equal in large degree to $B(E,\mathcal{M}(-\mathbb{d}'),\tau)$ need not be unique as subalgebras of $T$; determine whether they are unique up to isomorphism, or classify the isomorphism classes.

(D) Scope of the technical hypothesis: does Assumption 8.2 — existence of a minimal sporadic ideal for $T$ together with countability of $\operatorname{Div}(T\otimes_k K)$ over some uncountable algebraically closed extension — hold for further families satisfying Assumption 2.1, in particular for the Veronese rings of Stephenson's generic algebras associated to the elliptic curve?

(E) Genericity: Prove or refute the conjecture (Remark 6.10(iii)) that, generically, the blowup $T(\mathbb{d})$ has no sporadic ideals, so that the order and its maximal hull coincide ($U=F$ in Corollary 6.6).

References

1. D. Rogalski, S. J. Sierra, J. T. Stafford, Classifying orders in the Sklyanin algebra, Algebra & Number Theory 9 (2015), 2055-2119. DOI: 10.2140/ant.2015.9.2055, arXiv:1308.2213. https://arxiv.org/abs/1308.2213
2. 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://arxiv.org/abs/2008.06947
3. Dominic Hipwood, Maximal Order in the Sklyanin Algebra, PhD thesis, University of Manchester, 2018. arXiv:1812.04137. https://arxiv.org/abs/1812.04137
4. 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://arxiv.org/abs/1308.2216
5. D. Rogalski, Blowup subalgebras of the Sklyanin algebra, Advances in Mathematics 226 (2011), 1433-1473. DOI: 10.1016/j.aim.2010.08.009, arXiv:0912.2304. https://arxiv.org/abs/0912.2304
6. 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://arxiv.org/abs/1807.09889
7. Darin Stephenson, Algebras associated to elliptic curves, Transactions of the American Mathematical Society 349 (1997), 2317-2340. DOI: 10.1090/s0002-9947-97-01769-8. https://doi.org/10.1090/s0002-9947-97-01769-8
8. D. Rogalski, S. J. Sierra, J. T. Stafford, Ring-theoretic blowing down I, Journal of Noncommutative Geometry 11 (2017), 1465-1520. DOI: 10.4171/jncg/11-4-9, arXiv:1603.08128. https://arxiv.org/abs/1603.08128
9. D. Rogalski, S. J. 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://arxiv.org/abs/2107.01991