[Chow-Witt groups]
In this work we study the Chow-Witt groups. These groups were defined by J. Barge et F. Morel in order to understand when a projective module of top rank over a ring has a free factor of rank one, i.e., is isomorphic to . We show first that these groups satisfy the same functorial properties as the classical Chow groups. Then we define for each locally free -module of (constant) rank over a regular scheme an Euler class which is a refinement of the usual top Chern class . The Euler classes satisfy also good fonctorial properties. In particular, we get if is a projective module of rank over a regular ring of dimension such that . Next we compute the top Chow-Witt group of a regular ring of dimension and the top Chow-Witt group of a regular -algebra of finite dimension. For such , we get that if is a projective module of rank equal to the dimension of the ring then if and only if . Finally, we examine the links between the Chow-Witt groups and the Euler class groups defined by S. Bhatwadekar and R. Sridharan.
Dans ce travail, nous étudions les groupes de Chow-Witt. Ces groupes ont été introduits par J. Barge et F. Morel dans le but de comprendre dans quelle situation un -module projectif de rang égal à la dimension de est isomorphe à un module projectif plus simple . Dans un premier temps, nous montrons que ces groupes satisfont à peu de choses près les propriétés fonctorielles des groupes de Chow classiques. Nous définissons ensuite pour tout -module localement libre de rang (constant) sur un schéma régulier de dimension une classe d’Euler qui est un raffinement de la classe de Chern maximale classique . Cette classe d’Euler satisfait elle aussi de bonnes propriétés fonctorielles. Nous obtenons en particulier que si est un projectif de rang sur un anneau régulier de dimension supérieure ou égale à tel que alors . Nous calculons dans un second temps les groupes de Chow-Witt maximaux d’un anneau régulier de dimension et d’une -algèbre régulière de dimension quelconque. Il découle immédiatement de ces calculs que si est un -module projectif de rang égal à la dimension de l’anneau on a si et seulement si . Finalement nous examinons les liens entre les groupes de Chow-Witt et les groupes des classes d’Euler introduits par S. Bhatwadekar et R. Sridharan.
Mot clés : groupes de Chow-Witt, classe d’Euler, fibrés vectoriels
Keywords: Chow-Witt groups, Euler class, vector bundles
@book{MSMF_2008_2_113__1_0, author = {Fasel, Jean}, title = {Groupes de {Chow-Witt}}, series = {M\'emoires de la Soci\'et\'e Math\'ematique de France}, publisher = {Soci\'et\'e math\'ematique de France}, number = {113}, year = {2008}, doi = {10.24033/msmf.425}, mrnumber = {2542148}, zbl = {1190.14001}, language = {fr}, url = {http://www.numdam.org/item/MSMF_2008_2_113__1_0/} }
Fasel, Jean. Groupes de Chow-Witt. Mémoires de la Société Mathématique de France, Serie 2, no. 113 (2008), 205 p. doi : 10.24033/msmf.425. http://numdam.org/item/MSMF_2008_2_113__1_0/
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