Azumaya loci and discriminant ideals of PI algebras

Brown, K.A. and Yakimov, M.T. (2018) Azumaya loci and discriminant ideals of PI algebras. Advances in Mathematics, 340, pp. 1219-1255. (doi:10.1016/j.aim.2018.10.024)

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Abstract

We prove that, under mild assumptions, for all positive integers ℓ, the zero set of the discriminant ideal D ℓ (R/Z(R), tr)of a prime affine polynomial identity (PI) algebra R coincides with the zero set of the modified discriminant ideal MD ℓ (R/Z(R), tr) of R, and give an explicit description of this set in terms of the dimensions of the irreducible representations of R. Furthermore, we prove that, when ℓ is the square of the PI-degree of R, this zero set is precisely the complement of the Azumaya locus of R. This description is used to classify the Azumaya loci of the multiparameter quantized Weyl algebras at roots of unity. As another application, we prove that the zero set of the top discriminant ideal of a prime affine PI algebra R coincides with the singular locus of the center of R, provided that the discriminant ideal has height at least 2, R has finite global dimension and R is a Cohen–Macaulay module over its center.

Item Type:Articles
Additional Information:The research of KB was supported by Leverhulme Emeritus Fellowship EM-2017-081\9, and that of MY by NSF grant DMS-1601862 and Bulgarian Science Fund grant DN 02-5.
Keywords:Azumaya loci of PI algebras, discriminant ideals, algebras with trace, singular loci, Cohen–Macaulay modules, PI quantized Weyl algebras.
Status:Published
Refereed:Yes
Glasgow Author(s) Enlighten ID:Brown, Professor Ken
Authors: Brown, K.A., and Yakimov, M.T.
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Advances in Mathematics
Publisher:Elsevier
ISSN:0001-8708
ISSN (Online):1090-2082
Published Online:29 October 2018
Copyright Holders:Copyright © 2018 Elsevier Inc.
First Published:First published in Advances in Mathematics 340: 1219-1255
Publisher Policy:Reproduced in accordance with the publisher copyright policy
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Project CodeAward NoProject NamePrincipal InvestigatorFunder's NameFunder RefLead Dept
3008720Aspects of noncommutative geometry and noncommutative algebraKenneth BrownLeverhulme Trust (LEVERHUL)EM-2017-081\9M&S - Mathematics