## Noncommutative Auslander theorem

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- by Y.-H. Bao, J.-W. He and J. J. Zhang PDF
- Trans. Amer. Math. Soc.
**370**(2018), 8613-8638 Request permission

## 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.## References

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## Additional Information

**Y.-H. Bao**- Affiliation: School of Mathematical Sciences, Anhui University, Hefei, Anhui, 230601, People’s Republic of China
- MR Author ID: 873632
- Email: baoyh@ahu.edu.cn
**J.-W. He**- Affiliation: Department of Mathematics, Hangzhou Normal University, Hangzhou Zhejiang 310036, People’s Republic of China
- MR Author ID: 710882
- Email: jwhe@hznu.edu.cn
**J. J. Zhang**- Affiliation: Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195
- MR Author ID: 314509
- Email: zhang@math.washington.edu
- 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 ).
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**370**(2018), 8613-8638 - MSC (2010): Primary 16E65, 16E10
- DOI: https://doi.org/10.1090/tran/7332
- MathSciNet review: 3864389