Presentations of modules when ideals need not be principal
type: Item
instance of: scholarly article
title: Presentations of modules when ideals need not be principal
zbMATH Open document ID: 0641.16028
author: Robert Guralnick
author: Lawrence S. Levy
published in: Illinois Journal of Mathematics
publication date: 1988-01-01T00:00:00Z
review text: This paper studies presentations of modules. A presentation is a surjection \(f: P\to U\), where P is a finitely generated projective \(\Lambda\)-module and U is a \(\Lambda\)-module. Generalizing elementary divisor theory, two presentations f, g of U by P are called equivalent if \(f=\beta g\alpha\) for some \(\beta\) \(\in Aut U\) and \(\alpha\) \(\in Aut P\). If \(\Lambda\) is a commutative PID, one can restate the main result of elementary divisor theory as: any two presentations of U by P are equivalent. In this article, \(\Lambda\) is semiprime and module finite over its noetherian one dimensional center R. It is shown that the set of stable equivalence classes of presentations, pres(P,U), forms an abelian group. (Under mild hypotheses, stable equivalence classes are actual equivalence classes, for example, this is always true for \(\Lambda\) commutative.) If \(\Lambda\) is a prime maximal R-order, then \(pres(P,U)=0\) and if, in addition the uniform rank of the kernel is at least two, then any two presentations of U by P are equivalent. In particular, this answers a fifty year old question of Nakayama in the special case that \(\Lambda\) is a noncommutative PID module finite over its center. (The two authors, together with \textit{C. Odenthal} [in ``Elementary divisor theorem for noncommutative PID's'', Proc. Am. Math. Soc. (1988)] have completely settled the Nakayama question.) If \(\Lambda\) is not a maximal order, then ``unique presentation'' still holds in some situations. For example, this is true when \(\Lambda\) is a finitely generated algebra over an algebraically closed field of characteristic zero and Z has finite length. To study the problem when \(\Lambda\) is not a maximal order, a Mayer- Vietoris sequence is used. From this one finds that if I is a conductor ideal from a maximal order \(\Gamma\), P is a progenerator, and ker f is faithful, then pres(P,U) is a homomorphic image of \(K_ 1(\Lambda /I\Lambda)\). If \(\Lambda\) is commutative and U o…
Mathematics Subject Classification ID: 16Gxx
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16U30
Mathematics Subject Classification ID: 16Exx
Mathematics Subject Classification ID: 16D70
zbMATH DE Number: 4044035
zbMATH Keywords: presentations of modules
zbMATH Keywords: finitely generated projective \(\Lambda \) - module
zbMATH Keywords: semiprime
zbMATH Keywords: stable equivalence classes of presentations
zbMATH Keywords: noncommutative PID module finite over its center
zbMATH Keywords: prime maximal R-order
zbMATH Keywords: uniform rank
zbMATH Keywords: finitely generated algebra
zbMATH Keywords: Mayer-Vietoris sequence
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4044035