NONCOMMUTATIVE AUSLANDER THEOREM.

In the 1960s Maurice Auslander proved the following important result. Let R be the commutative polynomial ring C[x1, . . ., xn], and let G be a finite small subgroup of GLn(C) acting on R naturally. Let A be the fixed subring RG:= {a ∈ R|g(a) = a for all g ∈ G}. Then the endomorphism ring of the right A-module RA is naturally isomorphic to the skew group algebra R* 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. [ABSTRACT FROM AUTHOR]