Noncommutative Auslander theorem

Abstract

In the 1960s Maurice Auslander proved the following important result. Let $R$ be the commutative polynomial ring $\mathbb {C}[x_1,\dots ,x_n]$, and let $G$ be a finite small subgroup of $\textrm {GL}_n(\mathbb {C})$ acting on $R$ naturally. Let $A$ be the fixed subring $R^G:=\{a\in R|g(a)=a \text { for all } g\in G \}$. Then the endomorphism ring of the right $A$-module $R_A$ is naturally isomorphic to the skew group algebra $R\ast G$. In this paper, a version of the Auslander theorem is proven for the following classes of noncommutative algebras: (a) noetherian PI local (or connected graded) algebras of finite injective dimension, (b) universal enveloping algebras of finite-dimensional Lie algebras, and (c) noetherian graded down-up algebras.