**Memoires de la Societe Mathematique de France**

Volume: 113;
2008;
197 pp;
Softcover

MSC: Primary 13; 14; 18;
**Print ISBN: 978-2-85629-262-4
Product Code: SMFMEM/113**

List Price: $68.00

Individual Member Price: $54.40

# Groupes de Chow-Witt

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*Jean Fasel*

A publication of the Société Mathématique de France

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.

#### Readership

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