Groupes de Chow-Witt  [ Chow-Witt groups ]
Mémoires de la Société Mathématique de France, no. 113 (2008), 205 p.

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 QA. 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 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 c ˜ n (P)=0 if P is a projective module of rank n over a regular ring A of dimension n such that PQA. 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 -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 c ˜ n (P)=0 if and only if PQA. 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 QA. 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 mn une classe d’Euler 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 PQA alors 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 -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 c ˜ n (P)=0 si et seulement si PQA. 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.

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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