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Projective modules over polynomial rings: a constructive approach

✍ Scribed by S. Barhoumi; H. Lombardi; I. Yengui

Publisher: John Wiley and Sons
Year: 2009
Tongue: English
Weight: 138 KB
Volume: 282
Category: Article
ISSN: 0025-584X
DOI: 10.1002/mana.200610772

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✦ Synopsis

Abstract

We give a constructive proof of the fact that finitely generated projective modules over a polynomial ring with coefficients in a Prüfer domain R with Krull dimension ≤ 1 are extended from R. In particular, we obtain constructively that finitely generated projective R[X~1~, …, X~n~ ]‐modules, where R is a Bezout domain with Krull dimension ≤ 1, are free. Our proof is essentially based on a dynamical method for decreasing the Krull dimension and a constructive rereading of the original proof given by Maroscia and Brewer & Costa. Moreover, we obtain a simple constructive proof of a result due to Lequain and Simis stating that finitely generated modules over R[X~1~, …, X~n~ ], n ≥ 2, are extended from R if and only if this holds for n = 1, where R is an arithmetical ring with finite Krull dimension (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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