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Transactions of the American Mathematical Society

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Noncommutative Auslander theorem

Authors: Y.-H. Bao, J.-W. He and J. J. Zhang
Journal: Trans. Amer. Math. Soc. 370 (2018), 8613-8638
MSC (2010): Primary 16E65, 16E10
Published electronically: June 26, 2018
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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 $ {\rm GL}_n(\mathbb{C})$ acting on $ R$ naturally. Let $ A$ be the fixed subring $ R^G:=\{a\in R\vert 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.

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Additional Information

Y.-H. Bao
Affiliation: School of Mathematical Sciences, Anhui University, Hefei, Anhui, 230601, People’s Republic of China

J.-W. He
Affiliation: Department of Mathematics, Hangzhou Normal University, Hangzhou Zhejiang 310036, People’s Republic of China

J. J. Zhang
Affiliation: Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195

Keywords: Auslander theorem, Cohen-Macaulay property, Artin-Schelter regular algebra, pertinency, homologically small, Hopf algebra action
Received by editor(s): August 15, 2016
Received by editor(s) in revised form: May 20, 2017
Published electronically: June 26, 2018
Additional Notes: The first and second authors were supported by NSFC (grant Nos. 11571239, 11671351 and 11401001). The third author was supported by the US National Science Foundation (grant Nos. DMS-1402863 and DMS-1700825 ).
Article copyright: © Copyright 2018 American Mathematical Society

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