Artin–Schelter regular algebras
Let \(k\) be a field and let \(A = \bigoplus_{i \geq 0} A_i\) be a connected graded \(k\)-algebra, that is, \(A_0 = k\), and assume \(A\) is finitely generated. Write \(k = A/A_{\geq 1}\) for the trivial module.
The algebra \(A\) is Artin–Schelter regular (or AS-regular) of dimension \(d\) if:
- \(A\) has finite global dimension \(d\);
- \(A\) has finite Gelfand–Kirillov dimension, i.e. polynomial growth;
- \(A\) is Gorenstein: there is an integer \(\ell\) such that \[ \operatorname{Ext}^i_A(k, A) \cong \begin{cases} k(\ell) & i = d, \\ 0 & i \neq d. \end{cases} \]
These conditions were introduced by Artin–Schelter.
The algebra is quadratic if it is generated in degree \(1\) with all defining relations in degree \(2\). Every family on this site is quadratic — in fact Koszul, a stronger homological condition — so each is presented by four generators and six quadratic relations.
The basic example is the commutative polynomial ring \(k[x_1, \dots, x_d]\): it is AS-regular of dimension \(d\), and the noncommutative AS-regular algebras are exactly the noncommutative analogues of affine and projective space that this site is about. For \(d = 4\) the quadratic ones are the noncommutative \(\mathbb{P}^3\)’s.
Classification
In low dimensions the AS-regular algebras are completely understood.
- \(d = 1\). The only AS-regular algebra is the polynomial ring \(k[x]\).
- \(d = 2\). There are exactly two quadratic families on two generators: the quantum plane \(k\langle x, y\rangle/(yx - q\,xy)\) for \(q \in k^\times\), and the Jordan plane \(k\langle x, y\rangle/(yx - xy - x^2)\).
- \(d = 3\). The classification was begun by Artin–Schelter and completed, using the geometry of point schemes, by Artin–Tate–Van den Bergh. The generic algebras are governed by a triple \((E, \mathcal{L}, \sigma)\) of an elliptic curve with a line bundle and an automorphism; the quadratic case has three generators, the cubic case two.
In dimension \(d = 4\) no full classification is known. Restricting to algebras generated in degree \(1\), the minimal free resolution of the trivial module \(k\) takes one of three shapes, labelled by the ranks of its terms:
- \((1,4,6,4,1)\) — \(4\) generators and \(6\) relations of degree \(2\), with Hilbert series \(\operatorname{h}_A(t) = \dfrac{1}{(1-t)^4}\). These are exactly the quadratic algebras (automatically Koszul): the noncommutative \(\mathbb{P}^3\)’s, and the families catalogued on this site.
- \((1,3,4,3,1)\) — \(3\) generators and \(4\) relations, two of degree \(2\) and two of degree \(3\), with \(\operatorname{h}_A(t) = \dfrac{1}{(1-t)^3(1-t^2)}\).
- \((1,2,2,2,1)\) — \(2\) generators and \(2\) relations, one of degree \(3\) and one of degree \(4\), with \(\operatorname{h}_A(t) = \dfrac{1}{(1-t)^2(1-t^2)(1-t^3)}\).
The two non-quadratic types are far less understood. The \((1,2,2,2,1)\) algebras were largely classified by Lu–Palmieri–Wu–Zhang through the \(A_\infty\)-structure on their Ext-algebra (with the underlying machinery developed in a companion paper). Unlike the quadratic case, these two types have no classical geometric interpretation: the noncommutative projective scheme \(\operatorname{qgr} A\) is not \(\operatorname{coh} X\) for a smooth projective 3-fold \(X\), as discussed in Belmans, On non-quadratic 4-dimensional Artin–Schelter regular algebras and 3-folds.
Describing the known families of the quadratic type \((1,4,6,4,1)\) is the subject of this website.
Topics
- Cocycle twists — twisting by a finite group; how several families are obtained
- Double Ore extensions — the double Ore extensions of Zhang–Zhang, giving the 26 families A–Z
- Hochschild cohomology of qgr — deformations of the noncommutative variety
- Hochschild cohomology of the algebra — deformations of the graded algebra
- Line scheme — the scheme parametrising line modules
- Nakayama automorphism — the twist of the superpotential, and when the algebra is Calabi–Yau
- Noncommutative projective schemes — the qgr construction
- Normal elements — normal elements in low degrees
- Point scheme — the scheme parametrising point modules
- Superpotential — the relations as the derivatives of one element
- The centre — central elements and the centre's Hilbert series
- The Kodaira–Spencer map — how a family of algebras meets a component of the moduli stack