The genus of a module. II: Roiter's theorem, power cancellation and extension of scalars
type: Item
instance of: scholarly article
title: The genus of a module. II: Roiter's theorem, power cancellation and extension of scalars
zbMATH Open document ID: 0621.16007
DOI: 10.1016/0022-314X(87)90075-8
author: Robert Guralnick
published in: Journal of Number Theory
publication date: 1987-01-01T00:00:00Z
review text: In this interesting paper the author extends classical results on left \(\Lambda\)-lattices, where \(\Lambda\) is an order over the Dedekind domain \(R\) in a separable algebra over the quotient field \(K\) of \(R\), to the case of finitely generated left \(\Lambda\)-modules. More precisely, there exists an integer \(n=n(\Lambda)\), such that for any two finitely generated left \(\Lambda\)-modules \(M\) and \(N\), which are locally isomorphic, one has \(M^{(n)}\cong N^{(n)}\). If \(R\) satisfies the Jordan-Zassenhaus theorem, then \(n\) depends only on the skewfields in \(K\Lambda\) and the \(R\)-conductor from a maximal order \(\Gamma\supset \Lambda\). In the classical situation, this is Roiter's theorem. The important point is, that \(n\) can be chosen to be independent of \(M\) and \(N\), which is not so in general, if one considers finitely generated modules over a finitely generated \(R\)-algebra. The author then uses this to get generalizations of results on restricted genera, extension of scalars and cancellation to finitely generated \(\Lambda\)-modules. [Part I, cf. ibid. 18, 169--177 (1984; Zbl 0533.13006).]
reviewed by: Klaus W. Roggenkamp
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16U30
Mathematics Subject Classification ID: 16D80
zbMATH DE Number: 4006461
zbMATH Keywords: left \(\Lambda \)-lattices
zbMATH Keywords: order
zbMATH Keywords: separable algebra
zbMATH Keywords: finitely generated left \(\Lambda \)-modules
zbMATH Keywords: Jordan-Zassenhaus theorem
zbMATH Keywords: maximal order
zbMATH Keywords: Roiter's theorem
zbMATH Keywords: restricted genera
zbMATH Keywords: extension of scalars
zbMATH Keywords: cancellation
MaRDI profile type: Publication
cites work: Q5562621
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cites work: Module equivalences: Local to global. When primitive polynomials represent units
cites work: Krull-Schmidt and cancellation over local rings
cites work: The genus of a module
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cites work: Q3731730
cites work: Über die Geschlechter von Gittern über Ordnungen.
cites work: Genera and decompositions of lattices over orders
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description: scientific article; zbMATH DE number 4006461