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 \(A\) is finitely generated in degree \(1\). 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. Describing the known families 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
- Noncommutative projective schemes — the qgr construction
- Normal elements — normal elements in low degrees
- Point scheme — the scheme parametrising point modules
- 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