Maximal orders in the Sklyanin algebra

Abstract

A major current goal of noncommutative geometry is the classification of noncommutative projective surfaces. The generic case is to understand algebras birational to the Sklyanin algebra. In this work we complete a considerable component of this problem.

Let S denote the 3-dimensional Sklyanin algebra over an algebraically closed field, and assume that S is not a finite module over its centre. In earlier work Rogalski, Sierra and Stafford classified the maximal orders inside the 3-Veronese $S^{(3)}$ of S. We complete and extend their work and classify all maximal orders inside S. As in Rogalski, Sierra and Stafford's work, these can be viewed as blowups at (possibly non-effective) divisors. A consequence of this classification is that maximal orders are automatically noetherian among other desirable properties.

Introduction

Roughly, the goal of noncommutative projective geometry is to use techniques and intuition from commutative geometry to study noncommutative algebras. A major current goal of noncommutative geometry is the classification of noncommutative projective surfaces; in the language of algebras, a classification of connected graded domains of GK-dimension 3. Within this, a major target is a classification of algebras birational to the Sklyanin algebra. This problem provides the main motivation of our work.

First, as always, we need to get some essential definitions and formalities out of the way. Fix an algebraically closed field $\mathbb{k}$. Let $A={⨁}_{n\in \mathbb{N}}{A}_i$ be an $\mathbb{N}$-graded $\mathbb{k}$-algebra and a domain. We call A connected graded (cg) if $A_0=\mathbb{k}$ and ${\dim }_{\mathbb{k}}{A}_i<\infty$ for all i. Almost all algebras considered will be cg domains of finite Gelfand Kirillov dimension (see Notation 2.9 for a definition). In particular we assume this for the next few definitions.

The algebra A sits inside its graded quotient ring, $Q=Q_{\mathrm{gr}}(A)$, formed by inverting the Ore set of all nonzero homogeneous elements. The division ring of degree zero elements of Q will be denoted by $D_{\mathrm{gr}}(A)$. We call A a maximal order if there is no cg B with $A⊊B\subseteq Q$, and such that $xBy\subseteq A$ for some nonzero $x,y\in Q$. Let R be another algebra such that $A\subseteq R\subseteq Q$. We call A a a maximal R-order if there exists no graded equivalent order B to A such that $A⊊B\subseteq R$. Given $d\in \mathbb{N}$, the d-Veronese of A is the subring $A^{(d)}={⨁}_{i\in \mathbb{N}}{A}_{di}$. It is (usually) given the grading $A_n^{(d)}=A_{dn}$.

A construction that plays a fundamental role in our work, and in general noncommutative geometry, is that of a twisted homogeneous coordinate ring. Fix a projective scheme X. Let $\mathcal{L}$ be an invertible sheaf on X with global sections $H^0(X,\mathcal{L})$, and let $\sigma :X\to X$ be an automorphism of X. Write ${\mathcal{L}}^{\sigma }$ for the pullback sheaf ${\sigma }^{⁎}\mathcal{L}$. Set ${\mathcal{L}}_0={\mathcal{O}}_X$ and ${\mathcal{L}}_n=\mathcal{L}\otimes {\mathcal{L}}^{\sigma }\otimes \dots \otimes {\mathcal{L}}^{{\sigma }^{n-1}}$ for $n\ge 1$. We define $B(X,\mathcal{L},\sigma )={⨁}_{n\in \mathbb{N}}{H}^0(X,{\mathcal{L}}_n)$. There is a natural ring structure on $B(X,\mathcal{L},\sigma )$ and we call it a twisted homogeneous coordinate ring.

Our results regard certain subalgebras of the Sklyanin algebra, which we now define. Let $a,b,c\in \mathbb{k}$, then we set

$$S=\frac{\mathbb{k}〈x,y,z〉}{(azy+byz+c{x}^2,axz+bzx+c{y}^2,ayx+bxy+c{z}^2)}.$$

Provided $a,b,c$ are general enough, then S has a central element $g\in S_3$, unique up to scalar multiplication and such that $S/gS\cong B(E,\mathcal{L},\sigma )$. Here E is a nonsingular elliptic curve, $\mathcal{L}$ is an invertible sheaf of degree 3, and σ is an automorphism of E. We call such S a Sklyanin algebra.

The results of this work are analogous to those of [15], [17], where the authors tackled similar problems inside the 3-Veronese subring $T=S^{(3)}$ of S. Here, as in [15], [17], our results concern certain blowup subalgebras and virtual blowup subalgebras of S at effective and so-called virtually effective divisors on E. Our assumption throughout is Hypothesis 1.1.

Hypothesis 1.1 Standing assumption

Fix an algebraically closed field $\mathbb{k}$. Fix a 3 dimensional Sklyanin algebra S. Let $g\in S_3$ be such that $S/gS\cong B(E,\mathcal{L},\sigma )$, where E is a smooth elliptic curve, $\mathcal{L}$ an invertible sheaf on E with $\mathrm{deg}\mathcal{L}=3$ and $\sigma :E\to E$ an automorphism of E. Assume that σ is of infinite order.

Before presenting our results we set some notation as standard.

Notation 1.2

Given a subset $X\subseteq S$, we write $\bar{X}=(X+gS)/gS\subseteq S/gS$. Similarly for $x\in S$, $\bar{x}$ will denote its image in $\bar{S}=B(E,\mathcal{L},\sigma )$.

In [15] Rogalski studied what will be our 1 point blowup. The definition becomes slightly more complicated as we allow blowups at 2 points, since now the rings need not be generated in a single degree.

Definition 1.3 Definition 3.3 and Definition 3.7

Let d be an effective divisor on E with $\mathrm{deg}\mathbf{d}\le 2$. For $i=1,2,3$, put

• $V_1=\{x\in S_1|\bar{x}\in H^0(E,\mathcal{L}(-\mathbf{d}))\}$,

• $V_2=\{x\in S_2|\bar{x}\in H^0(E,{\mathcal{L}}_2(-\mathbf{d}-{\sigma }^{-1}(\mathbf{d})))\}$,

• $V_3=\{x\in S_3|\bar{x}\in H^0(E,{\mathcal{L}}_3(-\mathbf{d}-{\sigma }^{-1}(\mathbf{d})-{\sigma }^{-2}(\mathbf{d})))\}$.

We define the blowup of S at d to be the subalgebra of S generated by $V_1$, $V_2$ and $V_3$. We denote it by $S(\mathbf{d})=\mathbb{k}〈V_1,V_2,V_3〉$.

When $\mathbf{d}=p$ is a single point, we in fact have $S(p)=\mathbb{k}〈V_1〉$. Rogalski showed in [15] that it is the only degree 1 generated maximal order inside S. When $\mathbf{d}=p+q$ is two points Definition 1.3 is both new, and harder to understand than $S(p)$. The $S(\mathbf{d})$ are the correct analogue of the blowup subalgebras $S^{(3)}(\mathbf{e})$ of $S^{(3)}$ defined by Rogalski in [15, Section 1]. We now state the main results of this work.

Theorem 1.4 Theorem 3.15

Let d be an effective divisor on E of degree $d\le 2$. Set $R=S(\mathbf{d})$. Then:

• $R\cap gS=gR$ with $R/gR\cong B(E,\mathcal{L}(-\mathbf{d}),\sigma )$. The Hilbert series of R is

  $$h_R(t)=\sum _n({\dim }_{\mathbb{k}}{R}_n)t^n=\frac{t^2+(1-d)t+1}{{(t-1)}^2(1-t^3)}.$$

• The 3-Veronese $R^{(3)}$ is a blowup subalgebra of $S^{(3)}$. More specifically

  $$R^{(3)}=S^{(3)}(\mathbf{d}+{\sigma }^{-1}(\mathbf{d})+{\sigma }^{-2}(\mathbf{d})).$$

• R is a maximal order in $Q_{\mathrm{gr}}(R)=Q_{\mathrm{gr}}(S)$.

We also show in Theorem 3.15 that $S(\mathbf{d})$ satisfies some of the most useful homological properties, the most prominent for us will the Auslander-Gorenstein and Cohen-Macaulay properties (see Definition 3.1). Although not explicit below, obtaining Theorem 1.4 is absolutely essential for the rest of our main results.

For a complete classification of maximal S-orders we need to introduce virtual blowups at virtually effective divisors. We only define a virtually effective divisor here. For the purposes of the introduction one may take the conclusions of Proposition 1.6 as the definition of a virtual blowup.

Definition 1.5

A divisor x is called virtually effective if for all $n\gg 0$, the divisor $\mathbf{x}+{\sigma }^{-1}(\mathbf{x})+\dots +{\sigma }^{-(n-1)}(\mathbf{x})$ is effective.

Proposition 1.6 Proposition 4.22 and Definition 4.17

Let x be a virtually effective divisor of degree at most 2. Then there exists a virtual blowup $S(\mathbf{x})$. In particular:

• $S(\mathbf{x})\subseteq S_{(g)}$ where $S_{(g)}$ is the homogeneous localisation of S at the completely prime ideal gS.

• The ring $S(\mathbf{x})$ is a maximal order in $Q_{\mathrm{gr}}(S)$ and uniquely defines a maximal S-order $V=S(\mathbf{x})\cap S$.

• $S(\mathbf{x})\cap g{S}_{(g)}=gS(\mathbf{x})$ and, in high degrees $n\gg 0$,

  $${(S(\mathbf{x})/gS(\mathbf{x}))}_{\ge n}=B{(E,\mathcal{L}(-\mathbf{x}),\sigma )}_{\ge n}.$$

Despite the current notation, it is unknown whether the algebra $S(\mathbf{x})$ appearing in Proposition 1.6 is unique for a fixed virtually effective divisor x. Another unknown is when $S(\mathbf{x})\subseteq S$ holds. These problems are investigated in [9, Chapter 5.3].

The classification of maximal S-orders remarkably only requires blowups and virtual blowups at most 2 points.

Theorem 1.7 Theorem 5.7

Let U be a connected graded maximal S-order such that $\bar{U}\ne \mathbb{k}$. Then there exists a virtually effective divisor x with $0\le \mathrm{deg}\mathbf{x}\le 2$, and a virtual blowup $S(\mathbf{x})$, such that $S(\mathbf{x})$ is the unique maximal order containing $U=S(\mathbf{x})\cap S$.

It may not be the case that $S(\mathbf{x})\subseteq S$, however the difference between $S(\mathbf{x})$ and the ring $U=S(\mathbf{x})\cap S$ is small (see Theorem 5.8(1c)). Out of proving the above theorems we also obtain many nice properties for maximal orders. The most striking of these is that we get that maximal (S-)orders are noetherian for free. The homological terms in Corollary 1.8(2) will remain undefined for they are not used. The definitions can be found in [9, Chapter 2].

Corollary 1.8 Corollary 5.10 and Corollary 5.11

Let U be a cg maximal S-order such that $\bar{U}\ne \mathbb{k}$. Equivalently, let $U=S(\mathbf{x})\cap S$ for some virtual blowup $S(\mathbf{x})$ at a virtually effective divisor with $0\le \mathrm{deg}\mathbf{x}\le 2$. Then

• $S(\mathbf{x})$ and U are strongly noetherian, and are finitely generated as $\mathbb{k}$-algebras.

• $S(\mathbf{x})$ and U satisfy the Artin-Zhang χ conditions, have finite cohomological dimension, and possess balanced dualizing complexes.

The Auslander-Gorenstein and Cohen-Macaulay properties are missing from Corollary 1.8. It is shown in Theorem 6.1(3) that in general we cannot expect these homological properties.

The ultimate goal of this line of research is to understand connected graded algebras birational to the Sklyanin algebra. That is, algebras A satisfying $D_{\mathrm{gr}}(A)=D_{\mathrm{gr}}(S)$; or equivalently $Q_{\mathrm{gr}}(A)=Q_{\mathrm{gr}}{(S)}^{(d)}$ for some $d\ge 1$.

Theorem 1.9 Theorem 5.7

Let $d\ge 1$ be coprime to 3 and suppose that U is a cg maximal $S^{(d)}$-order satisfying $\bar{U}\ne \mathbb{k}$. Then there exists a virtually effective divisor x with $0\le \mathrm{deg}\mathbf{x}\le 2$, and virtual blowup $S(\mathbf{x})$, such that $U=S(\mathbf{x})\cap S^{(d)}$.

Retain the notation of Theorem 1.9. When $d=3e$ is divisible by 3, $U={(F\cap S)}^{(e)}$ for a virtual blowup F of $S^{(3)}$. This is [17, Theorem 8.11]. In contrast, when we prove Theorem 5.8 - the converse of Theorem 1.9 - we also prove the analogous statement for maximal $S^{(3)}$-orders. This result is an improvement on [17]. Out of these results we are able to obtain the best answer yet to [17, Question 9.4].

Corollary 1.10 Corollary 5.9

Let U be a cg graded subalgebra of S satisfying $D_{\mathrm{gr}}(U)=D_{\mathrm{gr}}(S)$ and such that $\bar{U}\ne \mathbb{k}$. If U is a maximal order then $U^{(d)}$ is a maximal order for all $d\ge 1$.

A final achievement of this work is an explicit construction of a virtual blowup - a first of its kind. The example shows that these algebras have certain intriguing and more technical properties, and the reader is referred to Section 6 for details.

In 1987, Artin and Schelter started a project to classify the noncommutative analogues of polynomial rings in 3 variables [2]. These algebras are now called AS-regular algebras. The subject of noncommutative projective geometry was born when Artin, Tate and Van den Bergh completed this classification in [5], [6]. Their results can be thought of as a classification of noncommutative projective planes.

More generally, one would like to classify all so-called noncommutative curves and surfaces. Let A be a cg noetherian domain, then we can associate a noncommutative projective scheme $\mathrm{qgr}(A)$ to A. It can be thought of as the noncommutative analogue of coherent sheaves, $\mathrm{coh}(X)$, over (the non-existent) $X=\mathrm{Proj}(A)$. The classification for noncommutative projective curves, when the GK-dimension is 2, was completed by Artin and Stafford in [4]. They show that in this case $\mathrm{qgr}(A)\sim \mathrm{coh}(X)$ for a genuine integral projective curve X. The question of noncommutative surfaces (when $\mathrm{GKdim}(A)=3$) is still very much open. It is this ultimate goal that motivates this work.

Let A be a cg domain with $\mathrm{GKdim}(A)=3$. Then $Q_{\mathrm{gr}}(A)=D_{\mathrm{gr}}(A)[t,t^{-1};\alpha ]$; a skew Laurent polynomial ring over the division ring $D_{\mathrm{gr}}(A)$ which is of transcendence degree 2. The division ring $D_{\mathrm{gr}}(A)$ is often called the noncommutative function field of A. A programme for the classification is to first classify the possible birational classes (the possible $D_{\mathrm{gr}}(A)$'s), and then classify the algebras in each birational equivalence class. Artin conjectures in [1] that we know all the possible division rings. They are:

• A division algebra which is finite dimensional over a central commutative subfield of transcendence degree 2.

• A division ring of factions of a skew polynomial extensions of $\mathbb{k}(X)$, for a commutative curve X.

• A noncommutative function field $D_{\mathrm{gr}}(S)$ of a 3 dimensional Sklyanin algebra S.

Whilst this conjecture is still a long way off, significant work has been, and is being, done on the classification of algebras in each birational class. Algebras with $D_{\mathrm{gr}}(A)$ commutative (plus a geometric condition) have been successfully classified by Rogalski and Stafford and then Sierra in [18], [19] and [20] respectively. This is a significant subclass of (1) above. We are interested in case (3) when $D_{\mathrm{gr}}(A)=D_{\mathrm{gr}}(S)$. More specifically, we look at subalgebras A of S with $D_{\mathrm{gr}}(A)=D_{\mathrm{gr}}(S)$. Where would be a good place to start looking for such algebras? Inside S of course! How about a target to aim for? Maximal orders are the noncommutative analogue of integrally closed domains, or geometrically, of normal varieties. They are therefore a natural target for such a classification.

The first major results in this direction were given by Rogalski. In [15] Rogalski classifies the degree 1 generated maximal orders of the 3-Veronese ring $T=S^{(3)}$ of S. These are classified as so-called blowup subalgebras $T(\mathbf{d})$ of T at effective divisors on E of degree at most 7. This is extended to include all maximal orders and maximal T-orders by Rogalski, Sierra and Stafford in [16], [17]. A detailed review of their work can be found in [9, Chapter 2.6-2.7]. In this work we ask the question, why work with $S^{(3)}$? Surely it is S we are interested in?!

Acknowledgements

This work showcases the main results of the author's PhD thesis. The author completed his PhD thesis at the University of Manchester under the supervision of Toby Stafford, and is extremely grateful for all the guidance provided by Toby through-out. The author would also like to thank EPSRC for the funding provided.