Algèbres de Lie de dimension infinie et théorie de la descente

Steinmetz Zikesch, Wilhelm Alexander

Mémoires de la Société Mathématique de France, no. 129 (2012) , 105 p.

Résumé
Abstract

Soit $k$ un corps algébriquement clos de caractéristique zéro et soit $R$ un anneau de polynômes de Laurent en deux variables sur $k$ . La motivation principale derrière ce travail est une classe d’algèbres de Lie de dimension infinie sur $k$ , appelées extended affine Lie algebras (EALAs). Ces algèbres correspondent à des torseurs sous des groupes algébriques linéaires sur $R$ . Dans ce travail nous classifions les $R$ -torseurs sous les groupes classiques de rang assez grand pour les types $A$ extérieur, $B, C, D$ et pour le type $A$ intérieur sous des hypothèses plus fortes. Ainsi, nous pouvons déduire des résultats sur des EALAs. Nous obtenons aussi une réponse affirmative à une variante de la conjecture II de Serre pour l’anneau $R$ : tout $R$ -torseur lisse sous un groupe semi-simple simplement connexe de rang assez grand de type classique $B$ , $C$ et $D$ est trivial.

Let $k$ be an algebraically closed field of characteristic zero and let $R$ be the Laurent polynomial ring in two variables over $k$ . The main motivation behind this work is a class of infinite dimensional Lie algebras over $k$ , called extended affine Lie algebras (EALAs). These algebras correspond to torsors under algebraic groups over $R$ . In this work we classify $R$ -torsors under classical groups of large enough rank for outer type $A$ and types $B, C, D$ , as well as for inner type $A$ under stronger hypotheses. We can thus deduce results on EALAs. We also obtain a positive answer to a variant of Serre’s Conjecture II for the ring $R$ : every smooth $R$ -torsor under a semi-simple simply connected $R$ -group of large enough rank of classical type $B, C, D$ is trivial.

MR Zbl

DOI : 10.24033/msmf.440

Mot clés : Algèbre de Lie de dimension infinie, EALA, anneau de polynômes de Laurent, cohomologie galoisienne, groupe algébrique linéaire classique, algèbre d’Azumaya à involution, forme hermitienne, théorie de Witt triangulaire
Mots clés : Infinite Dimensional Lie Algebra, Extended Affine Lie Algebra (EALA), Laurent Polynomial Ring, Galois Cohomology, Classical Linear Algebraic Group, Azumaya Algebra with Involution, Hermitian Form, Triangular Witt Theory

@book{MSMF_2012_2_129__1_0,
     author = {Steinmetz Zikesch, Wilhelm Alexander},
     title = {Alg\`ebres de {Lie} de dimension infinie et th\'eorie de la descente},
     series = {M\'emoires de la Soci\'et\'e Math\'ematique de France},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {129},
     year = {2012},
     doi = {10.24033/msmf.440},
     mrnumber = {3059229},
     zbl = {1319.17014},
     language = {fr},
     url = {http://www.numdam.org/item/MSMF_2012_2_129__1_0/}
}

TY  - BOOK
AU  - Steinmetz Zikesch, Wilhelm Alexander
TI  - Algèbres de Lie de dimension infinie et théorie de la descente
T3  - Mémoires de la Société Mathématique de France
PY  - 2012
IS  - 129
PB  - Société mathématique de France
UR  - http://www.numdam.org/item/MSMF_2012_2_129__1_0/
DO  - 10.24033/msmf.440
LA  - fr
ID  - MSMF_2012_2_129__1_0
ER  -

%0 Book
%A Steinmetz Zikesch, Wilhelm Alexander
%T Algèbres de Lie de dimension infinie et théorie de la descente
%S Mémoires de la Société Mathématique de France
%D 2012
%N 129
%I Société mathématique de France
%U http://www.numdam.org/item/MSMF_2012_2_129__1_0/
%R 10.24033/msmf.440
%G fr
%F MSMF_2012_2_129__1_0

Steinmetz Zikesch, Wilhelm Alexander. Algèbres de Lie de dimension infinie et théorie de la descente. Mémoires de la Société Mathématique de France, Série 2, no. 129 (2012), 105 p. doi : 10.24033/msmf.440. http://numdam.org/item/MSMF_2012_2_129__1_0/

Bibliographie
Cité par

[1] Allison (Bruce), Azem (Saeid), Berman (Stephen), Gao (Yun) & Pianzola (Arturo) – Extended affine Lie algebras and their root systems, Mem. Am. Math. Soc., t.603 (1997), pp.1–122. | MR

[2] Allison (Bruce), Berman (Stephen), Faulkner (John) & Pianzola (Arturo) – Realization of graded-simple algebras as loop algebras, Forum Math. (à paraître) (2007). | MR

[3] Bak (Anthony) – On modules with quadratic forms, in “Algebraic K-theory and its Geometric applications", Conf. Hull, Lecture Notes in Math., vol.108, Springer Verlag, 1969, pp.55–66. | MR

[4] Balmer (Paul) – Triangular Witt groups. Part I : The 12-term localisation exact sequence, K-theory, t.19 (2000), pp.311–363. | MR

[5] —, Triangular Witt groups. Part II : From usual to derived, Math. Z., t.236 (2001), pp.351–382. | MR | Zbl

[6] Balmer (Paul) & Preeti (Raman) – Shifted Witt groups of semi-local rings, Manuscripta Math., t.117 (2005), pp.1–27. | MR

[7] Balmer (Paul) & Walter (Charles) – A Gersten-Witt Spectral Sequence For Regular Schemes, Ann. Sci. École Norm. Sup., $4^{e}$ sér., t.35 (2002), pp.127–152. | MR | EuDML

[8] Bass (Hyman) – On the ubiquity of Gorenstein rings, Math. Z., t.82 (1963), pp.8–28. | MR | EuDML | Zbl

[9] —, Algebraic k-theory, W.A.Benjamin, Inc., 1968.

[10] Bass (Hyman), Kuku (Aderemi O.) & Pedrini (Claudio) – Proceedings of the Workshop and Symposium “Algebraic K-Theory and its Applications", World Scientific, 1999. | MR | Zbl

[11] Bayer-Fluckiger (Eva) & Parimala (Raman) – Galois cohomology of the classical groups over fields of cohomological dimension $\leq 2$ , Invent. Math., t.122 (1995), pp.195–229. | MR | EuDML

[12] Borel (Armand) – A Linear Algebraic Groups, 2nd ed., Graduate Texts in Math., vol.126, Springer-Verlag, New York, 1991. | MR

[13] Borel (Armand) & Tits (Jacques) – Groupes réductifs, Pub. Math. IHÉS, t.27 (1965), pp.55–152. | MR | EuDML

[14] Bruns (Winfried) & Herzog (Jürgen) – Cohen Macaulay Rings, Cambridge Univ. Press, 1993. | MR

[15] Demazure (M.) & Grothendieck (A.) – Séminaire de géométrie algébrique de l’IHÉS, 1963–1964, schémas en groupes, Lecture Notes in Math., vol.151-153, Springer-Verlag, 1970. | MR

[16] Eisenbud (David) – Commutative algebra with a view toward algebraic geometry, Graduate Texts in Math., Springer Verlag, 1994. | MR

[17] Fulton (William) – Intersection Theory, 2nd ed., Ergeb. Math. Grenzgeb., vol.3, 1998. | MR

[18] Gabber (Ofer) – Some theorems on Azumaya algebras, Lecture Notes in Math., vol.844, pp.129–209. | MR

[19] Gille (Philippe) & Pianzola (Arturo) – Torsors, Reductive Group Schemes and Extended Affine Lie Algebras.

[20] —, Isotriviality of torsors over Laurent polynomial rings, C. R. Acad. Sci. Paris, Sér. I, t.340 (2005), pp.725–729. | Zbl

[21] —, Galois cohomology and forms of algebras over Laurent polynomial rings, Math. Ann., t.338 (2007), pp.497–543. | MR | Zbl

[22] —, Isotriviality and étale cohomology of Laurent polynomial rings, J. Pure. Appl. Alg., t.212 (2008), pp.780–800. | MR | Zbl

[23] Gille (Philippe) & Szamuely (Tamás) – Central Simple Algebras and Galois Cohomology, Cambridge Studies in Advanced Math., vol.101, Cambridge University Press, 2006. | MR

[24] Gille (Stefan) – On Witt groups with support, Math. Ann., t.322 (2002), pp.103–137. | MR

[25] —, Homotopy invariance of coherent Witt groups, Math. Z., t.244 (2003), pp.211–233. | MR | Zbl

[26] —, A transfer morphism for Witt groups, J. reine angew. Math., t.564 (2003), pp.215–233. | MR | Zbl

[27] —, A Gersten-Witt Complex For Hermitian Witt Groups of Coherent Algebras Over Schemes I : Involutions Of The First Kind, Compos. Math., t.143 (2007), pp.271–289. | MR | Zbl

[28] —, The general dévissage theorem for Witt groups of schemes, Arch. Math., t.88 (2007), pp.333–343. | MR | Zbl

[29] —, A graded Gersten-Witt complex for schemes with a dualizing complex and the Chow group, J. Pure Appl. Alg., t.208 (2007), pp.391–419. | MR | Zbl

[30] —, A Gersten-Witt Complex For Hermitian Witt Groups of Coherent Algebras Over Schemes II : Involutions Of The Second Kind, J. K-Theory, t.4 (2009), pp.347–377. | MR | Zbl

[31] Giraud (Jean) – Cohomologie non abélienne, Grundlehren Math. Wiss., vol.179, Springer-Verlag, Berlin, 1971. | MR

[32] Grothendieck (Alexander) – Le groupe de Brauer I, in “Dix exposés sur la cohomologie des schémas", North-Holland, Amsterdam, Masson, Paris, 1968. | MR

[33] —, Le groupe de Brauer II, Théories cohomologiques, in “Dix exposés sur la cohomologie des schémas", North-Holland, Amsterdam, Masson, Paris, 1968. | Zbl

[34] Hartshorne (Robin) – Residues and Duality, Lecture Notes in Mathematics, vol.20, Springer-Verlag, 1966. | MR | EuDML

[35] De Jong (Aise J.) – The period-index problem for the Brauer group of an algebraic surface, Duke Math. J., t.123 (2004), pp.71–94. | MR | Zbl

[36] Kneser (Martin) – On Galois Cohomology, Lecture Notes, Tata Institute of Fundamental Research, Bombay, 1969.

[37] Knus (Max-Albert) – Algèbres d’Azumaya et modules projectifs, Comm. Math. Helv. (1969). | MR

[38] —, Quadratic and Hermitian Forms over Rings, Grundlehren Math. Wiss., vol.294, Springer-Verlag, 1991. | Zbl

[39] Knus (Max-Albert), Merkurjev (Alexander), Tignol (Jean-Pierre) & Rost (Markus) – The Book of Involutions, AMS Colloquium Publications, vol.44, 1998. | MR

[40] Lam (Tsit Yuen) – Serre’s Problem On Projective Modules, Springer Mathematical Monographs, 2006. | MR

[41] Matsumura (Hideyuki) – Commutative Ring Theory, Cambridge Studies, vol.8, Cambridge University Press, 1986. | MR

[42] Merkurjev (Alexander) – Invariants of algebraic groups, J. reine angew. Math., t.508 (1999), pp.127–156. | MR

[43] Milne (James S.) – Étale Cohomology, Princeton University Press, 1980. | MR

[44] Milnor (John Willard) – Introduction to Algebraic K-theory, Princeton University Press, 1972. | MR

[45] Neher (Erhard) – Extended affine Lie algebras, C. R. Math. Rep. Acad. Sci. Canada, t.26 (2004), pp.90–96. | MR

[46] Panin (Ivan A.) & Suslin (Andrei A.) – On a Grothendieck Conjecture for Azumaya Algebras, St. Petersburg Math. J., t.9 (1998), pp.851–858. | MR

[47] Parimala (Raman) – Quadratic Spaces Over Laurent Extensions of Dedekind Domains, Trans. Amer. Math. Soc., t.277 (1983), pp.569–578. | MR | Zbl

[48] Pianzola (Arturo) – Vanishing of $H^{1}$ for Dedekind rings and application to loop algebras, Comptes Rendus, Acad, Sci. Paris, sér. I, t.340 (2005), pp.633–638. | MR | Zbl

[49] Quebbemann (Heinz-Georg), Scharlau (Winfried) & Schulte (Manfred) – Quadratic and Hermitian forms in additive and abelian categories, J. Algebra, t.59 (1979), pp.264–289. | MR

[50] Rosenlicht (Maxwell) – Toroidal algebraic groups, Proc. Amer. Math. Soc., t.12 (1961), pp.984–988. | MR | Zbl

[51] Sansuc (Jean-Jacques) – Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres, J. reine angew. Math., t.327 (1981), pp.12–80. | MR | EuDML | Zbl

[52] Scharlau (Winfried) – Quadratic and Hermitian Forms, Grundlehren Math. Wiss., vol.270, Springer-Verlag, 1985. | MR

[53] Serre (Jean-Pierre) – Cohomologie Galoisienne, 5 $^{e}$ éd., Springer-Verlag, 1997. | MR

[54] Thélène (Jean-Louis Colliot), Gille (Philippe) & Parimala (Raman) – Arithmetic of linear algebraic groups over two-dimensional geometric fields, Duke Math. J., t.121 (2004), pp.285–321. | MR

[55] Tits (Jacques) – Classification of Algebraic Semisimple Groups, Algebraic Groups and Discontinous Subgroups, in “Proceedings of Symposia in Pure Mathematics IX", Boulder, Colorado 1965, American Math. Soc., 1966, pp.33–62. | MR

[56] Weil (André) – Algebras with involution and the classical groups, J. Ind. Math. Soc., t.24 (1961), pp.589–623. | MR

Cité par Sources :