Projective modules over real affine algebras
type: Item
instance of: scholarly article
title: Projective modules over real affine algebras
zbMATH Open document ID: 0696.13007
DOI: 10.1007/BF01446885
author: Manuel Ojanguren
author: R. Parimala
published in: Mathematische Annalen
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164677
review text: For any field k, all projective modules over the polynomial ring \(R=k[X_ 1,...,X_ d]\) are free, by the celebrated theorem of Quillen and Suslin. On the other hand, if \(d\geq 3\) and \(f\in R\), then all projective modules over the ring \(k[X_ 1,...,X_ d,1/f]\) need not be free [see example 1 in \textit{R. G. Swan's} paper in Trans. Am. Math. Soc. 105, 264-277 (1962; Zbl 0109.416)]. It is proved that if \(k={\mathbb{R}}\) and f is strictly positive then all projective modules over \({\mathbb{R}}[X_ 1,X_ 2,X_ 3,1/f]\) are free. This fact is a consequence of the following theorem: Let R be a real closed field and A an affine domain over R of dimension d, smooth in codimension one. Let \(f\in A\) be a function without any R- rational zeros and P a projective module of rank \(\geq d-1\) over \(A_ f\). If \(P\oplus A_ f\) is free then P is extended from A.
Mathematics Subject Classification ID: 13C10
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 14Pxx
Mathematics Subject Classification ID: 14A05
zbMATH DE Number: 4139899
zbMATH Keywords: freeness of projective modules over polynomial rings
zbMATH Keywords: Rees ring
zbMATH Keywords: real closed field
zbMATH Keywords: rank
reviewed by: Patrick F. Smith
MaRDI profile type: Publication
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OpenAlex ID: W2061189057
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description: scientific article; zbMATH DE number 4139899