Groupes de Chow-Witt - 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 | 1 citation in Numdam

DOI: 10.24033/msmf.425

Classification: 13C10, 13D15, 14C15, 14C17, 18F30
Keywords: Chow-Witt groups, Euler class, vector bundles

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     title = {Groupes de {Chow-Witt}},
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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/

References
Cited by

[Ba1] P. Balmer, Derived Witt groups of a scheme, Journal of pure and applied alg. 141, no. 2 (1999), 101-129. | Zbl | MR

[Ba2] P. Balmer, Triangular Witt Groups. Part I : the 12-term exact sequence, K-theory 19 (2000), 311-363. | Zbl | MR

[Ba3] P. Balmer, Triangular Witt Groups. Part II : from usual to derived, Math. Zeitschrift 236, no. 2 (2001), 351-382. | Zbl | MR

[Ba4] P. Balmer, Products of degenerate quadratic forms, Compositio Math. 141 (2005), 1374-1404. | Zbl | MR

[BG] P. Balmer, S. Gille, Koszul complexes and symmetric forms over the punctured affine space, Proceedings of the London Mathematical Society 91, no 2 (2005), 273-299. | Zbl | MR

[BW] P. Balmer, C. Walter, A Gersten-Witt spectral sequence for regular schemes, Annales Scientifiques de l’ENS. 35, no. 1 (2002), 127-152. | Numdam | Zbl | EuDML | MR

[BM] J. Barge, F. Morel, Groupe de Chow des cycles orientés et classe d’Euler des fibrés vectoriels, C. R. Acad. Sci. Paris 330 (2000), 287-290. | MR

[BO] J. Barge, M. Ojanguren, Fibrés algébriques sur une surface réelle, Comment. Math. Helv. 62 (1987), 616-629. | Zbl | EuDML | MR

[BOU] N. Bourbaki, Algèbre, Eléments de mathématiques, chapitre 10, Masson (1980), Paris. | Zbl | MR

[BS1] S. M. Bhatwadekar, Raja Sridharan, Projective generation of curves in polynomial extensions of an affine domain and a question of Nori, Invent. Math. 133 (1998), 161-192. | Zbl | MR

[BS2] S. M. Bhatwadekar, Raja Sridharan, Zero cycles and the Euler class groups of smooth real affine variety, Invent. Math. 136 (1999), 287-322. | Zbl | MR

[BS3] S. M. Bhatwadekar, Raja Sridharan, The Euler class group of a noetherian ring, Compositio Math. 122 (2000), 183-222. | Zbl | MR

[BH] W. Bruns, J. Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, 39. Cambridge University Press, Cambridge, 1993. | Zbl | MR

[Fa] J. Fasel, The Chow-Witt ring, Doc. Math. 12 (2007), 275–312. | Zbl | EuDML | MR

[Fu] W. Fulton, Intersection theory, Ergebnisse der Math. und ihrer Grenzgebiete (1984), Springer. | Zbl | MR

[Gi1] S. Gille, On Witt-groups with support, Thèse, 2001.

[Gi2] S. Gille, A transfer morphism for Witt groups, J. reine angew. Math. 564 (2003), 215-233. | Zbl | MR

[Gi3] S. Gille, A graded Gersten-Witt complex for schemes with dualizing complex and the Chow group, J. Pure Appl. Algebra 208 (2007), no. 2, 391–419. | Zbl | MR

[Gi4] S. Gille, The general dévissage theorem for Witt groups of schemes Arch. Math. (Basel) 88 (2007), no. 4, 333–343. | Zbl | MR

[Gro1] A. Grothendieck, Eléments de géométrie algébrique (EGA) II, Publ. math IHES 32 (1967).

[Gro2] A. Grothendieck, Eléments de géométrie algébrique (EGA) III, Publ. math IHES 32 (1967).

[Gro3] A. Grothendieck, Eléments de géométrie algébrique (EGA) IV, Publ. math IHES 32 (1967). | Zbl

[Ha1] R. Hartshorne, Residues and duality, Lecture Notes in Math. 20 (1966), Springer. | EuDML | MR

[Ha2] R Hartshorne, Algebraic geometry, Graduate text in Math. 52 (1977), Springer. | MR

[Kn] M. Knebusch, On algebraic curves over real closed fields I, Math. Zeitschrift 150 (1976), 49-70. | Zbl | EuDML | MR

[Ku] E. Kunz, Kähler differentials, Adv. lectures in Math. (1986), Vieweg und Sohn. | Zbl | MR

[Ma] B. Magurn, An algebraic introduction to K-theory, Encyclopedia of Math. and its Applic. 87 (2002). | Zbl | MR

[Mat] H. Matsumura, Commutative ring theory, Cambridge Studies in Adv. Math., Cambridge University Press (1986), Cambridge.

[Mi] J. Milnor, Algebraic K-theory and quadratic forms, Invent. Math. 9 (1969-1970), 318-344. | EuDML | MR

[MH] J. Milnor, D. Husemoller, Symmetric bilinear forms, Ergebnisse der Math. und ihrer Grenzgebiete 73 (1973), Springer. | Zbl | MR

[Mo] F. Morel, $𝔸^{1}$ -homotopy classification of vector bundles over smooth affine schemes, preprint available at http ://www.mathematik.uni-muenchen.de/ morel/preprint.html

[Mu1] M. P. Murthy, A survey of obstruction theory for projective modules of top rank, Contemp. Math. 243 (1999), 153-174. | Zbl | MR

[Mu2] M. P. Murthy, Zero cycles and projective modules, Ann. Math. 140 (1994), 405-434. | Zbl | MR

[Pl] B. R. Plumstead, The conjectures of Eisenbud and Evans, Amer. Journal of Math. 105 (1983), 1417-1433. | Zbl | MR

[QSS] H.-G. Quebbemann, W. Scharlau, M.Schulte, Quadratic and hermitian forms in additive and abelian categories, Journal of Algebra 59, no. 2 (1979), 264-290. | Zbl | MR

[Ro] M. Rost, Chow groups with coefficients, Doc. Math. 1 (1996), 319-193. | Zbl | EuDML | MR

[Sc] W. Scharlau, Quadratic and hermitian forms, Grundlehren der math. Wissen. 270 (1985), Springer. | Zbl | MR

[Sch] M. Schmid, Wittringhomologie, Thèse, 1997.

[Se] J.-P. Serre, Corps locaux, Public. institut math. université Nancago VIII (1968), Hermann. | MR

[Sw] R. G. Swan, A cancellation theorem for projective modules in the metastable range, Invent. Math. 122 (1985), 113-153. | EuDML | MR

[Vo] V. Voevodsky, Motivic cohomology with $ℤ / 2$ coefficients, Public. Hautes Etudes Sci. 98 (2003), 59-104. | Numdam | Zbl | EuDML | MR

[We] C. A. Weibel, An introduction to homological algebra, Cambridge Studies in Adv. Math., Cambridge University Press (1994), Cambridge. | Zbl | MR

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