Class groups of maximal orders over Krull domains
type: Item
instance of: scholarly article
title: Class groups of maximal orders over Krull domains
zbMATH Open document ID: 0535.16005
DOI: 10.1016/0022-4049(84)90080-X
published in: Journal of Pure and Applied Algebra
publication date: 1984-01-01T00:00:00Z
review text: Let R be a (commutative) Krull domain with field of fractions K and let \(\Lambda\) be a maximal order over R in some central simple algebra \(\Sigma\) over K. The normalizing classgroup of \(\Lambda\), \(C\ell(\Lambda)\), is studied by geometric tools. (\(C\ell(\Lambda)\) is expressed in terms of Picard groups of certain varieties.) To this end Weil and Cartier divisors are introduced in the case of ''noncommutative affine schemes'' associated with \(\Lambda\). Since all maximal orders are PI-rings this can be done by extending divisors of the central scheme. The methods are applied to obtain the following results: - a characterization of those locally factorial Krull domains R for which all maximal orders in \(M_ n(K)\) are conjugated, - the property that every torsion element of the Picard group of R is killed in the class group of a certain \(M_ n(R),\) - a proof that \(C\ell(\Lambda)\) is a direct summand of \(C\ell(\Lambda t).\) In the last section some results of the central classgroup of \(\Lambda\) are given. Here the approach is more ring theoretical.
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16U30
Mathematics Subject Classification ID: 16P10
Mathematics Subject Classification ID: 16Rxx
Mathematics Subject Classification ID: 16E20
Mathematics Subject Classification ID: 14A20
zbMATH DE Number: 3849419
zbMATH Keywords: maximal order
zbMATH Keywords: central simple algebra
zbMATH Keywords: normalizing classgroup
zbMATH Keywords: Picard groups
zbMATH Keywords: noncommutative affine schemes
zbMATH Keywords: PI-rings
zbMATH Keywords: locally factorial Krull domains
author: Lieven Le Bruyn
reviewed by: Jan Van Geel
MaRDI profile type: Publication
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full work available at URL: 0022-4049(84)90080-x
OpenAlex ID: W2078573757
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description: scientific article; zbMATH DE number 3849419