Volume (2008) no. 113

Groupes de Chow-Witt [ Chow-Witt groups ]

Fasel, Jean

Mémoires de la Société Mathématique de France, no. 113 (2008) , 205 p.

Abstract
Résumé

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 $P$ of top rank over a ring $A$ has a free factor of rank one, i.e., is isomorphic to $Q \oplus A$ . We show first that these groups satisfy the same functorial properties as the classical Chow groups. Then we define for each locally free $𝒪_{X}$ -module $E$ of (constant) rank $n$ over a regular scheme $X$ an Euler class ${\tilde{c}}_{n} (E)$ which is a refinement of the usual top Chern class $c_{n} (E)$ . The Euler classes satisfy also good fonctorial properties. In particular, we get ${\tilde{c}}_{n} (P) = 0$ if $P$ is a projective module of rank $n$ over a regular ring $A$ of dimension $n$ such that $P ≃ Q \oplus A$ . Next we compute the top Chow-Witt group of a regular ring $A$ of dimension $2$ and the top Chow-Witt group of a regular $ℝ$ -algebra $A$ of finite dimension. For such $A$ , we get 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 $P ≃ Q \oplus A$ . 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 $A$ -module projectif $P$ de rang égal à la dimension de $A$ est isomorphe à un module projectif plus simple $Q \oplus A$ . 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 $𝒪_{X}$ -module localement libre $E$ de rang (constant) $n$ sur un schéma régulier $X$ de dimension $m \geq n$ une classe d’Euler ${\tilde{c}}_{n} (E)$ qui est un raffinement de la classe de Chern maximale classique $c_{n} (E)$ . Cette classe d’Euler satisfait elle aussi de bonnes propriétés fonctorielles. Nous obtenons en particulier que si $P$ est un projectif de rang $n$ sur un anneau régulier $A$ de dimension supérieure ou égale à $n$ tel que $P ≃ Q \oplus A$ alors ${\tilde{c}}_{n} (P) = 0$ . Nous calculons dans un second temps les groupes de Chow-Witt maximaux d’un anneau régulier de dimension $2$ et d’une $ℝ$ -algèbre $A$ régulière de dimension quelconque. Il découle immédiatement de ces calculs que si $P$ est un $A$ -module projectif de rang $n$ égal à la dimension de l’anneau on a ${\tilde{c}}_{n} (P) = 0$ si et seulement si $P ≃ Q \oplus A$ . 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.

MR 2542148 | Zbl 1190.14001

DOI : https://doi.org/10.24033/msmf.425
Classification: 13C10, 13D15, 14C15, 14C17, 18F30
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},
     zbl = {1190.14001},
     mrnumber = {2542148},
     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://www.numdam.org/item/MSMF_2008_2_113__1_0/

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