Azumaya algebras which are not smash products
type: Item
instance of: scholarly article
title: Azumaya algebras which are not smash products
zbMATH Open document ID: 0705.16009
author: Lindsay N. Childs
published in: Rocky Mountain Journal of Mathematics
publication date: 1990-01-01T00:00:00Z
review text: Let R be a commutative ring, H and R-Hopf algebra, finitely-generated and projective as R-module, \(H^*\) the dual Hopf algebra. Let Gal(H) be the set of isomorphism classes (as R-algebras and H-modules) of Galois H- extensions. There is a smash-product map {\#}: Gal(H)\(\times G(H^*)\to Az(R)\) to the set of isomorphism classes of Azumaya R-algebras. Over number fields, every Azumaya algebra is isomorphic to such a smash product. This paper indicates the failure of this result over number rings. Thus, let R be the ring of integers in a number field K. For each R-Hopf algebra of rank 2, {\#} maps into \(Az_ 2(R)\), the set of isomorphism classes of rank 4 Azumaya R-algebras. The author's main theorem is as follows: Let n be an even integer. With at most finitely- many exceptions, for every totally real number field K of dimension n over the rationals, there exists an element of \(Az_ 2(R)\) which is not representable as a smash-product. This theorem applies to real quadratic fields. For \(K=Q(\sqrt{p})\), p a prime, he obtains the number of isomorphism classes of Azumaya R-algebras of rank 4 which are smash- products (in all cases, it is \(\leq 3)\), and he shows that there are rank 4-Azumaya algebras which are not smash-products for all \(p>3\).
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 11R54
Mathematics Subject Classification ID: 11R04
Mathematics Subject Classification ID: 16S30
Mathematics Subject Classification ID: 16W30
zbMATH DE Number: 4156677
zbMATH Keywords: dual Hopf algebra
zbMATH Keywords: Galois H-extensions
zbMATH Keywords: smash-product
zbMATH Keywords: Azumaya R- algebras
zbMATH Keywords: number rings
zbMATH Keywords: ring of integers
zbMATH Keywords: totally real number field
zbMATH Keywords: real quadratic fields
zbMATH Keywords: rank 4-Azumaya algebras
reviewed by: Earl J. Taft
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181073160
OpenAlex ID: W2090837100
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DOI: 10.1216/RMJM/1181073160
description: scientific article; zbMATH DE number 4156677