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Groupes de Chow-Witt
Jean Fasel, ETH Zentrum, Zurich, Switzerland
A publication of the Société Mathématique de France.
Mémoires de la Société Mathématique de France
2008; 197 pp; softcover
Number: 113
ISBN-10: 2-85629-262-3
ISBN-13: 978-2-85629-262-4
List Price: US$68
Individual Members: US$61.20
Order Code: SMFMEM/113
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In this work the author studies the Chow-Witt groups. These groups were defined by J. Barge and F. Morel in order to understand when a projective module \(P\) of top rank over a ring \(A\) has a free factor of rank one, i.e., is isomorphic to \(Q\oplus A\).

First the author shows that these groups satisfy the same functorial properties as the classical Chow groups. Then he defines for each locally free \(\mathcal O_X\)-module \(E\) of (constant) rank \(n\) over a regular scheme \(X\) an Euler class \(\tilde{c}_n(E)\) that is a refinement of the usual top Chern class \(c_n(E)\). The Euler classes also satisfy good functorial properties. In particular, \(\tilde{c}_n(P)=0\) if \(P\) is a projective module of rank \(n\) over a regular ring \(A\) of dimension \(n\) such that \(P\simeq Q\oplus A\).

Next the author computes the top Chow-Witt group of a regular ring \(A\) of dimension \(2\) and the top Chow-Witt group of a regular \(\mathbb R\)-algebra \(A\) of finite dimension. For such \(A\), he obtains that if \(P\) is a projective module of rank equal to the dimension of the ring then \(\tilde{c}_n(P)=0\) if and only if \(P\simeq Q\oplus A\).

Finally, the author examines the links between the Chow-Witt groups and the Euler class groups defined by S. Bhatwadekar and R. Sridharan.

A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.


Graduate students and research mathematicians interested in algebra and algrebraic geometry.

Table of Contents

  • Introduction
  • Le complexe en \(K\)-théorie de Milnor
  • Le complexe de Gersten-Witt d'un schéma régulier
  • Le complexe de Gersten-Witt d'un schéma de Gorenstein
  • Le morphisme de transfert
  • Le calcul du morphisme de transfert
  • Un autre calcul des différentielles du complexe
  • Le morphisme de transfert pour les morphismes propres
  • Complexe de Gersten-Witt et idéaux fondamentaux
  • Groupes de Chow-Witt d'un schéma
  • Invariances homotopiques
  • Produits fibrés et morphismes de complexes
  • Les classes d'Euler
  • La classe d'Euler d'un module projectif de rang maximal
  • La dimension 2
  • Le groupe de Chow-Witt maximal d'une \(\mathbb{R}\)-algèbre lisse
  • Les groupes des classes d'Euler
  • Théorème d'Eisenbud-Evans et Théorème de Bertini
  • Catégories triangulées
  • Le groupe de Witt d'une catégorie exacte
  • Les groupes de Witt de catégories traingulées
  • Remarques sur les groupes de Witt d'un corps
  • Bibliographie
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