The Bass-Murthy question: Serre dimension of Laurent polynomial extensions
type: Item
instance of: scholarly article
title: The Bass-Murthy question: Serre dimension of Laurent polynomial extensions
zbMATH Open document ID: 0604.13007
DOI: 10.1007/BF01388777
published in: Inventiones Mathematicae
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143252
review text: The Serre dimension of a commutative ring R is the smallest t such that every projective R-module is a direct sum of a free module and a module of rank \(\leq t.\) This paper is an important contribution to the study of the relation between the Krull dimension of R and the Serre dimension of Laurent and polynomial extensions of R. The main result is that the Serre dimension is independent of the number and type of variables: Theorem: Let R be a noetherian commutative ring of Krull dimension d, and let \(A=R[X_ 1,...,X_ n,Y_ 1^{\epsilon_ 1},...,Y_ m^{\epsilon_ m}]\) where \(\epsilon_ i=\pm 1\). Then every projective A-module of rank \(\geq d+1\) has a unimodular element. The algebraic technique employed is to obtain a 2-cover of spec(A), on each piece of which the projective module has a unimodular element, and then to patch these unimodular elements by a split automorphism.
Mathematics Subject Classification ID: 13C15
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13F25
zbMATH DE Number: 3975062
zbMATH Keywords: Laurent extensions
zbMATH Keywords: Serre dimension
zbMATH Keywords: Krull dimension
zbMATH Keywords: polynomial extensions
zbMATH Keywords: unimodular element
author: S. M. Bhatwadekar
author: Hartmut Lindel
author: Ravi A. Rao
reviewed by: Phillip Schultz
MaRDI profile type: Publication
cites work: Grothendieck groups and Picard groups of abelian group rings
cites work: Some theorems about projective modules over polynomial rings
cites work: Generating modules efficiently: Theorems from algebraic K-theory
cites work: Generation and projective generation of ideals
cites work: Projective modules over some polynomial rings
cites work: Localisation de formes quadratiques $I$
cites work: Q3825170
cites work: On efficient generation of ideals
cites work: The Conjectures of Eisenbud and Evans
cites work: Projective modules over polynomial rings
cites work: Stability theorems for overrings of polynomial rings. II
cites work: Application of patching diagrams to some questions about projective modules
cites work: ON THE STRUCTURE OF THE SPECIAL LINEAR GROUP OVER POLYNOMIAL RINGS
cites work: Projective Modules over Laurent Polynomial Rings
cites work: The general linear group of polynomial rings over regular rings
OpenAlex ID: W1994620730
Recommended article: A note on the Serre dimension of polynomial rings
Recommended article: An extended form of the Grothendieck-Serre conjecture
Recommended article: On Bass' question for finitely generated algebras over large fields
Recommended article: Buchsbaum and Monomial Conjecture Dimension
Recommended article: A theorem on smoothness- Bass-Quillen, Chow groups and intersection multiplicity of Serre
Recommended article: Serre dimension and Euler class groups of overrings of polynomial rings
Recommended article: The dimension conjecture for polydisc algebras
Recommended article: On the Gorenstein injective dimension and Bass formula
Recommended article: Crystalline extensions and the weight part of Serre's conjecture
description: scientific article; zbMATH DE number 3975062