Noncommutative Auslander theorem. (English) Zbl 1432.16011
Summary: 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 \( \mathrm{GL}_n(\mathbb{C})\) 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.
MSC:
| 16E65 | Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.) |
| 16S50 | Endomorphism rings; matrix rings |
| 16E10 | Homological dimension in associative algebras |
Keywords:
Auslander theorem; Cohen-Macaulay property; Artin-Schelter regular algebra; pertinency; homologically small; Hopf algebra actionReferences:
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