Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-05T10:40:04.112Z Has data issue: false hasContentIssue false

L'invariant de Suslin en caractéristique positive

Published online by Cambridge University Press:  08 June 2010

Tim Wouters
Affiliation:
K.U.Leuven, Departement Wiskunde, Celestijnenlaan 200B bus 2400, B-3001 Leuven, Belgique, [email protected]
Get access

Abstract

For a central simple k-algebra A with indk (A) ∈ k×, Suslin defined a cohomological invariant for SK1 (A) [Sus2]. In this text, we generalise his invariant to any central simple k-algebra using a lift from positive characteristic to characteristic 0. To be able to define the invariant, we use Kato's cohomology of logarithmic differentials [Kat1].

Résumé

Pour une k-algèbre simple centrale A d'indice inversible dans k, Suslin a défini un invariant cohomologique de SK1 (A) ‘Sus2’. Dans ce texte, nous généralisons cet invariant à toute k-algèbre simple centrale par un relèvement de la caractéristique positive à la caractéristique 0. Pour pouvoir définir cet invariant, on a besoin des groupes de cohomologie des différentielles logarithmiques de Kato [Kat1].

Type
Research Article
Copyright
Copyright © ISOPP 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Références

AG.Auslander, Maurice et Goldman, Oscar. The Brauer group of a commutative ring. Trans. Amer. Math. Soc. 97, 367409, 1960.Google Scholar
Alb.Albert, Adrian. Simple algebras of degree pe over a centrum of characteristic p. Trans. Amer. Math. Soc. 40(1), 112126, 1936.Google Scholar
BK.Bloch, Spencer et Kato, Kazuya. p-adic étale cohomology. Publ. Math. Inst. Hautes Études Sci. (63), 107152, 1986.Google Scholar
Bla.Blanchet, Altha. Function fields of generalized Brauer-Severi varieties. Comm. Algebra 19(1), 97118, 1991.Google Scholar
BT.Bass, Hyman et Tate, John. The Milnor ring of a global field. In Algebraic K-theory, II : “Classical” algebraic K-theory and connections with arithmetic (Proc. Conf., Seattle, Wash., Battelle Memorial Inst., 1972), pages 349446. Lecture Notes in Math. 342. Springer, Berlin, 1973.Google Scholar
Car.Cartier, Pierre. Questions de rationalité des diviseurs en géométrie algébrique. Bull. Soc. Math. France 86, 177251, 1958.CrossRefGoogle Scholar
Coh.Cohen, Irvin. On the structure and ideal theory of complete local rings. Trans. Amer. Math. Soc. 59, 54106, 1946.CrossRefGoogle Scholar
EKLV.Esnault, Hélène, Kahn, Bruno, Levine, Marc, et Viehweg, Eckart. The Arason invariant and mod 2 algebraic cycles. J. Amer. Math. Soc. 11(1), 73118, 1998.Google Scholar
Gil1.Gille, Philippe. Invariants cohomologiques de Rost en caractéristique positive. K-Theory 21, 57100, 2000.Google Scholar
Gil2.Gille, Philippe. Le problème de Kneser-Tits. Astérisque 326, 2009. Séminaire Bourbaki 983.Google Scholar
GMS.Garibaldi, Skip, Merkurjev, Alexander, et Serre, Jean-Pierre. Cohomological invariants in Galois cohomology, University Lecture Series 28, Amer. Math. Soc., 2003.Google Scholar
Gro1.Grothendieck, Alexander. Éléments de Géométrie Algébrique IV, Étude locale des schémas et des morphismes de schémas Première Partie 20 de Publ. Math. Inst. Hautes Études Sci., Bures-sur-Yvette, 1964.CrossRefGoogle Scholar
Gro2.Grothendieck, Alexander. Le groupe de Brauer : I. Algèbres d'Azumaya et interprétations diverses. Séminaire Bourbaki 9, 199219, 19641966. Exposé No. 290.Google Scholar
GS.Gille, Philippe et Szamuely, Tamás. Central Simple Algebras and Galois Cohomology, Cambridge studies in advanced mathematics 101, Cambridge University Press, Cambridge, 2006.CrossRefGoogle Scholar
Har.Hartshorne, Robin. Algebraic Geometry, Graduate Texts in Mathematics 52, Springer Science+Business Media, Inc., New York, 1977.CrossRefGoogle Scholar
Izh.Izhboldin, Oleg. On the cohomology groups of the field of rational functions. In Mathematics in St. Petersburg, Amer. Math. Soc. Transl. Ser. 2 174, 2144, 1996.Google Scholar
Kah1.Kahn, Bruno. Applications of weight-two motivic cohomology. Doc Math. J. DMV 1, 395416, 1996.Google Scholar
Kah2.Kahn, Bruno. Cohomological approaches to SK1 and SK2 of central simple algebras. Preprint, 2009.Google Scholar
Kat1.Kato, Kazuya. Galois cohomology of complete discrete valuation fields. In Algebraic K-Theory, Lecture notes in mathematics 967, 215238, Springer-Verlag, Berlin, 1982.CrossRefGoogle Scholar
Kat2.Kato, Kazuya. A Hasse principle for two-dimensional global fields. J. Reine Angew. Math. 366, 142183, 1986.Google Scholar
KMRT.Knus, Max-Albert, Merkurjev, Alexander, Rost, Markus, et Tignol, Jean-Pierre. The book of involutions, Amer. Math. Soc. Colloq. Publ. 44, 1998.Google Scholar
Knu.Knus, Max-Albert. Quadratic and Hermitian forms over rings, Grundlehren der Mathematischen Wissenschaften 294, Springer-Verlag, Berlin, 1991.CrossRefGoogle Scholar
Lic.Lichtenbaum, Stephen. The construction of weight-two arithmetic cohomology. Invent. math. 88, 183215, 1987.Google Scholar
Mat.Matsumura, Hideyuki. Commutative ring theory, Cambridge Studies in Advanced Mathematics 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid.Google Scholar
Mer1.Merkurjev, Alexander. Generic element in SK 1 for simple algebras. K-Theory, 7(1), 13, 1993.Google Scholar
Mer2.Merkurjev, Alexander. Invariants of algebraic groups. J. reine angew. Math. 508, 127156, 1999.Google Scholar
Mil1.Milne, James. Algebraic number theory, 2009. http://www.jmilne.org/math/.Google Scholar
Mil2.Milnor, John. Algebraic K-theory and quadratic forms. Invent. Math. 9, 318344, 1969/1970.CrossRefGoogle Scholar
NM.Nakayama, Tadasi et Matsushima, Yozô. Über die multiplikative Gruppe einer p-adischen Divisionsalgebra. Proc. Imp. Acad. Tokyo 19, 622628, 1943.Google Scholar
Pan.Panin, Ivan. Splitting principle and K-theory of simply connected semisimple algebraic groups. Algebra i Analiz 10(1), 88131, 1998.Google Scholar
Pla.Platonov, Vladimir. The Tannaka-Artin problem and reduced K-theory. Math. USSR Izv. 10(2), 211243, 1976. Traduction anglaise.Google Scholar
Ros1.Rost, Markus. Chow Groups with Coefficients. Doc. Math. J. DMV 1, 319393, 1996.Google Scholar
Ros2.Rost, Markus. The basic correspondence of a splitting variety. Notes téléchargables de son site personnel, 1998.Google Scholar
RTY.Rehman, Ulf, Tikhonov, Sergey, et Yanchevskiĭ, Vyacheslav. Symbols and cyclicity of algebras after a scalar extension. Preprint, 2008.Google Scholar
San.Sansuc, Jean-Jacques. Groupe de Brauer et arithmétique des groupes algébriques linéaires. J. reine angew. Math. 327, 1280, 1981.Google Scholar
Sch.Schoeller, Colette. Groupes affines, commutatifs, unipotents sur un corps parfait. Bulletin de la S.M.F. 100, 241300, 1972.Google Scholar
Ser1.Serre, Jean-Pierre. Corps Locaux. Publications de l'Institut de Mathématique de l'Université de Nancago. Hermann, Paris, 1968.Google Scholar
Ser2.Serre, Jean-Pierre. Galois Cohomology. Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2002.Google Scholar
Sus1.Suslin, Andrei. SK1 of division algebras and Galois cohomology. In Algebraic K-theory, Adv. Soviet Math. 4, 7599. Amer. Math. Soc., Providence, RI, 1991.Google Scholar
Sus2.Suslin, Andrei. SK1 of division algebras and Galois cohomology revisited. Math. Soc. Transl. Ser. 2, 219, 125147. Am. Math. Soc., 2006.Google Scholar
Voe.Voevodsky, Vladimir. On Motivic Cohomology with ℤ/l coefficients. Prépublication, 2009.Google Scholar
Wad.Wadsworth, Adrian. Valuation theory on finite dimensional division algebras. In Valuation theory and its applications, Vol. I (Saskatoon, SK, 1999), Fields Inst. Commun. 32, 385449. Amer. Math. Soc., Providence, RI, 2002.Google Scholar
Wan.Wang, Shianghaw. On the commutator group of a simple algebra. Amer. J. Math. 72, 323334, 1950.Google Scholar
Wei.Weibel, Charles. The norm residue isomorphism theorem. J. Topology 2, 346372, 2009.Google Scholar
Wit.Witt, Ernst. Zyklische Körper und Algebren der Charakteristic p vom Grad pn. J. reine angew. Math. 176, 126140, 1937.CrossRefGoogle Scholar
Wou.Wouters, Tim. Comparing invariants of SK1. Prépublication, 2010.Google Scholar