Unimodular elements in projective modules
type: Item
instance of: scholarly article
title: Unimodular elements in projective modules
zbMATH Open document ID: 0839.13006
DOI: 10.1016/S0021-8693(05)80004-3
author: Hartmut Lindel
published in: Journal of Algebra
publication date: 1996-06-19T00:00:00Z
review text: Finitely generated modules over Noetherian commutative rings of finite Krull dimension are considered. An approach to establish the existence of unimodular elements in modules over positively graded rings and certain localizations of these rings are developed. In the case of a projective module \(P\) over the Laurent polynomial ring \(A = R [X_1, \dots, X_n\), \(Y_1^{\pm 1}, \dots, Y_m^{\pm 1}]\), the author obtains results corresponding to some results by \textit{S. M. Bhatwadekar}, \textit{H. Lindel} and \textit{R. A. Rao} [Invent. Math. 81, 189-203 (1985; Zbl 0604.13007)] without using the Quillen-Suslin theorem on projective modules over polynomial rings. The transitivity of the action of the group of elementary transformations on the set of unimodular elements \(\text{Um} (A \oplus P)\) in the case \(\text{rank} P \geq \max (2, \dim R + 1)\) is proved. A direct proof of the corresponding result by \textit{A. Wiemers} [``Eigenschaften projektiver Moduln über Laurent-Polynomringen und diskreten Hodge-Algebren'', Diss. (Münster 1989; Zbl 0716.13007)] concerning large projective modules over discrete Hodge algebras \(R[X_1, \dots, X_n]/I\), where \(I\) is generated by monomials or \(I = 0\), is given.
Mathematics Subject Classification ID: 13C10
Mathematics Subject Classification ID: 13F25
Mathematics Subject Classification ID: 13F50
zbMATH DE Number: 763195
zbMATH Keywords: graded rings
zbMATH Keywords: projective module
zbMATH Keywords: Laurent polynomial ring
zbMATH Keywords: unimodular elements
zbMATH Keywords: Hodge algebras
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 763195