Witt morphisms
type: Item
instance of: scholarly article
zbMATH Open document ID: 1257.11039
author: Przemysław Koprowski
publication date: 2012-08-08T00:00:00Z
Mathematics Subject Classification ID: 11E81
Mathematics Subject Classification ID: 11E10
Mathematics Subject Classification ID: 11-02
Mathematics Subject Classification ID: 14H05
zbMATH DE Number: 6064120
zbMATH Keywords: Witt ring
zbMATH Keywords: Witt functor
zbMATH Keywords: Witt equivalence
zbMATH Keywords: real closed field
zbMATH Keywords: function field of a curve
zbMATH Keywords: Scharlau transfer
zbMATH Keywords: Scharlau's norm principle
zbMATH Keywords: Knebusch-Milnor exact sequence
zbMATH Keywords: real holomorphy ring
MaRDI profile type: Publication
title: Witt morphisms
published in: Prace Naukowe Uniwersytetu Śląskiego w Katowicach
review text: The present book is a thorough presentation of the theory of function fields of curves over a real closed field, their Witt rings and Witt equivalence between such fields, where fields are called Witt equivalent if their Witt rings are isomorphic, and injectivity and surjectivity questions when passing from a Witt ring of a domain to that of certain ring extensions. There are many related aspects that are also presented in detail such as Witt equivalence of real holomorphy rings. Some of the sections consist of work published earlier by the author, some contain new material.NEWLINENEWLINENEWLINESection 1 gives an introduction to the basic notions and relevant known results such as Witt rings, orderings and valuations, and the Knebusch-Milnor exact sequence of Witt rings NEWLINE\[NEWLINE0\to WP\to WK\to \bigoplus WK(\mathfrak{p})NEWLINE\]NEWLINE where \(P\) is a Dedekind domain, \(K\) its quotient field, \(K(\mathfrak{p})\) the residue field at a maximal ideal \(\mathfrak{p}\), and where the sum ranges over all maximal ideals of \(P\). A short introduction to real curves is also included. In the situation where \(K\) is the function field of a real curve \(\gamma\) over a real closed field and \(R\) is the ring of regular functions on \(\gamma\), it is shown how the Knebusch-Milnor exact sequence extends to Knebusch's exact sequence NEWLINE\[NEWLINE0\to WR\to WK\to \bigoplus WK(\mathfrak{p})\to\mathbb{Z}^N\to 0NEWLINE\]NEWLINE where \(N\) is the number of semi-algebraically connected components of \(\gamma\). In Section 4, this exact sequence is revisited and it is shown that it splits, more precisely, that there is an exact sequence going from right to left in with the maps are suitably compatible with the ones in the original Knebusch sequence, i.e. in homological language, that the Knebusch sequence slices into and is patched by two split short exact sequences. It is furthermore shown that if \(P\) is the ring of polynomial functions on \(\gamma\) and if \(\gamma\…
reviewed by: Detlev Hoffmann
description: scientific article; zbMATH DE number 6064120