nLab Artin-Schelter regular algebra

Idea

A useful smoothness criterium for noncommutative graded algebras.

Definition

Let $k$ be a field and $A$ a connected finitely generated associative $\mathbb{N}$-graded $k$-algebra.

$A$ is Artin-Schelter regular (AS regular) if the following three conditions hold:

1. it is of finite global dimension $d$

2. has polynomial growth ($dim A_n$ is bounded above by a polynomial function $f(n)$)

3. is Gorenstein: $Ext_A^i({}_A k,A) = 0$ for $i\neq d$ and $Ext_A^d({}_A k,A) = k_A$.

Literature

The notion was introduced (under the name regular algebra) in :

• Michael Artin, W. F. Schelter, Graded algebras of global dimension 3, Adv. Math. 66 2 (1987) 171-216 [doi:10.1016/0001-8708(87)90034-X]

category: algebra