Picard groups of Witt rings
type: Item
instance of: scholarly article
title: Picard groups of Witt rings
zbMATH Open document ID: 0702.11024
DOI: 10.1007/BF02571345
author: Robert W. Fitzgerald
published in: Mathematische Zeitschrift
publication date: 1991-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/174227
review text: For an arbitrary Witt ring R it is shown that the Picard group Pic(R) is isomorphic to the ideal class group. Exact sequences of K-theory then may be used to improve earlier results on the ideal class group (which required R to have only finitely many orderings). Analysis of one such sequence shows that if s is the reduced stability index of R and e is the exponent of Pic(R) then \(e=\max \{1,2^{s-2}\}\) whenever e or s is finite. In particular \(Pic(R)=1\) (that is, all invertible ideals are principal) if and only if \(s\leq 2\). A complete computation of Pic(R) is also given for R with only finitely many orderings. This involves K- theory and Marshall's classification of finitely generated reduced Witt rings.
Mathematics Subject Classification ID: 11E81
Mathematics Subject Classification ID: 19G12
Mathematics Subject Classification ID: 19G05
zbMATH DE Number: 4150277
zbMATH Keywords: Witt ring
zbMATH Keywords: Picard group
zbMATH Keywords: K-theory
zbMATH Keywords: reduced stability index
zbMATH Keywords: finitely many orderings
MaRDI profile type: Publication
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OpenAlex ID: W1994007758
description: scientific article; zbMATH DE number 4150277