Formes quadratiques et cycles algébriques  [ Quadratic forms and algebraic cycles ]
Séminaire Bourbaki : volume 2004/2005, exposés 938-951, Astérisque no. 307  (2006), Talk no. 941, p. 113-163

The theory of quadratic forms over a field was introduced by Witt in 1937. It plays a key rôle in Voevodsky’s proofs of the Milnor conjectures via the pioneering work of Rost. Conversely, the methods of Rost and Voevodsky using the theory of motives and motivic Steenrod operations have had a revolutionary impact on the theory of quadratic forms and have led to proofs of basic results that seemed previously inaccessible. We shall explain, among other things, how these methods yield a proof that, if q is an anisotropic form in I n (the n-th power of the augmentation ideal in the Witt ring) and dimq<2 n+1 , then dimq is of the form 2 n+1 -2 i for some integer i{0,,n}.

Introduite par Witt en 1937, la théorie des formes quadratiques sur un corps joue un rôle central dans la démonstration des conjectures de Milnor par Voevodsky via les travaux pionniers de Rost qui y interviennent. Réciproquement, les méthodes de Rost et Voevodsky utilisant la théorie des motifs et les opérations de Steenrod motiviques révolutionnent la théorie des formes quadratiques et ont conduit à la démonstration de résultats de base qui semblaient auparavant inaccessibles. On expliquera notamment comment ces méthodes permettent de démontrer que, si q est une forme quadratique anisotrope dans I n (puissance n-ième de l’idéal d’augmentation de l’anneau de Witt) et que dimq<2 n+1 , alors dimq est de la forme 2 n+1 -2 i pour un entier i{0,,n}.

Classification:  11E04,  14C25
Keywords: quadratic forms, algebraic cycles, motives
@incollection{SB_2004-2005__47__113_0,
     author = {Kahn, Bruno},
     title = {Formes quadratiques et cycles alg\'ebriques},
     booktitle = {S\'eminaire Bourbaki : volume 2004/2005, expos\'es 938-951},
     author = {Collectif},
     series = {Ast\'erisque},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {307},
     year = {2006},
     note = {talk:941},
     pages = {113-163},
     zbl = {1120.11018},
     mrnumber = {2296417},
     language = {fr},
     url = {http://www.numdam.org/item/SB_2004-2005__47__113_0}
}
Kahn, Bruno. Formes quadratiques et cycles algébriques, in Séminaire Bourbaki : volume 2004/2005, exposés 938-951, Astérisque, no. 307 (2006), Talk no. 941, pp. 113-163. http://www.numdam.org/item/SB_2004-2005__47__113_0/

[1] Y. André - “Motifs de dimension finie (d'après Kimura, O'Sullivan...)”, in Séminaire Bourbaki (2003/04), Astérisque, vol. 299, Paris, 2005, Exp. 929, p. 115-145. | Numdam | MR 2167204 | Zbl 1080.14010

[2] Y. André & B. Kahn - “Nilpotence, radicaux et structures monoïdales (avec un appendice de P. O'Sullivan)”, Rend. Sem. Mat. Univ. Padova 108 (2002), p. 107-291. | Numdam | MR 1956434 | Zbl 1165.18300

[3] -, “Erratum : Nilpotence, radicaux et structures monoïdales”, Rend. Sem. Mat. Univ. Padova 113 (2005), p. 125-128. | Numdam | MR 2168984 | Zbl 1167.18302

[4] J. Kr. Arason - “Cohomologische Invarianten quadratischer Formen”, J. Algebra 36 (1975), p. 448-491. | MR 389761 | Zbl 0314.12104

[5] J. Kr. Arason & M. Knebusch - “Über die Grade quadratischer Formen”, Math. Ann. 234 (1978), p. 167-192. | MR 506027 | Zbl 0362.10021

[6] J. Kr. Arason & A. Pfister - “Beweis des Krullschen Durchschnittsatzes für den Wittring”, Invent. Math. 12 (1971), p. 173-176. | MR 294251 | Zbl 0212.37302

[7] P. Balmer - “Witt cohomology, Mayer-Vietoris, Homotopy invariance and the Gersten Conjecture”, K-Theory 23 (2001), p. 15-30. | MR 1852452 | Zbl 0987.19002

[8] P. Balmer & C. Walter - “A Gersten-Witt spectral sequence for regular schemes” 35 (2002), p. 127-152. | Numdam | MR 1886007 | Zbl 1012.19003

[9] A. Borel - Linear algebraic groups, 2e 'ed., Springer, 1991. | Article | MR 1102012 | Zbl 0726.20030

[10] P. Brosnan - “A short proof of Rost nilpotence via refined correspondences”, 8 (2003), p. 69-78. | MR 2029161 | Zbl 1044.11017

[11] -, “Steenrod operations in Chow theory”, Trans. Amer. Math. Soc. 355 (2003), p. 1869-1903. | Article | MR 1953530 | Zbl 1045.55005

[12] V. Chernousov, S. Gille & A. Merkurjev - “Motivic decomposition of isotropic projective homogeneous varieties”, Duke Math. J. 126 (2005), p. 137-159. | Article | MR 2110630 | Zbl 1086.14041

[13] C. Chevalley - “Sur les décompositions cellulaires des espaces G/B, in Algebraic groups and their generalizations : classical methods, Proc. Sympos. Pure Math., vol. 56 (I), Providence, RI, 1994, p. 1-23. | MR 1278698 | Zbl 0824.14042

[14] M. Demazure - “Motifs de variétés algébriques”, in Séminaire Bourbaki (1969/70), Lect. Notes in Math., vol. 180, Springer, 1971, Exp. 365, p. 19-38. | Numdam | Zbl 0247.14004

[15] L. E. Dickson - “On quadratic forms in a general field”, 14 (1907), p. 108-115, Math. papers IV, Chelsea, 1975, p. 512-519. | JFM 38.0182.02 | MR 1558550

[16] D. Edidin & W. Graham - “Equivariant intersection theory”, Invent. Math. 131 (1998), p. 595-634. | Article | MR 1614555 | Zbl 0940.14003

[17] R. W. Fitzgerald - “Function fields of quadratic forms”, Math. Z. 178 (1981), p. 63-76. | Article | MR 627094 | Zbl 0442.10013

[18] E. Friedlander - “Motivic complexes of Suslin-Voevodsky”, in Séminaire Bourbaki (1996/97), Astérisque, vol. 245, Paris, 1997, Exp. 833, p. 355-378. | Numdam | MR 1627118 | Zbl 0908.19005

[19] W. Fulton - Intersection theory, 2e 'ed., Springer, 1984, 1998. | Article | MR 1644323 | Zbl 0885.14002

[20] D. K. Harrison - “A Grothendieck ring of higher degree forms”, J. Algebra 35 (1975), p. 123-138. | Article | MR 379370 | Zbl 0309.15016

[21] D. W. Hoffmann - “Isotropy of quadratic forms over the function field of a quadric”, Math. Z. 220 (1995), p. 461-476. | Article | MR 1362256 | Zbl 0840.11017

[22] -, “Splitting patterns and invariants of quadratic forms”, Math. Nachr. 190 (1998), p. 149-168. | Article | MR 1611608 | Zbl 0916.11022

[23] J. Hurrelbrink & U. Rehmann - “Splitting patterns of quadratic forms”, Math. Nachr. 176 (1995), p. 111-127. | Article | MR 1361129 | Zbl 0876.11017

[24] O. Izhboldin - “Motivic equivalence of quadratic forms”, 3 (1998), p. 341-351. | MR 1668530 | Zbl 0957.11019

[25] -, “Fields of u-invariant 9, Ann. of Math. (2) 154 (2001), p. 529-587. | MR 1884616 | Zbl 0998.11015

[26] O. Izhboldin & A. Vishik - “Quadratic forms with absolutely maximal splitting”, in Quadratic forms and their applications (Dublin, 1999), Contemp. Math., vol. 272, Providence, RI, 2000, p. 103-125. | MR 1803363 | Zbl 0972.11017

[27] U. Jannsen - “Motives, numerical equivalence and semi-simplicity”, Invent. Math. 107 (1992), p. 447-452. | Article | MR 1150598 | Zbl 0762.14003

[28] B. Kahn - “The total Stiefel-Whitney class of a regular representation”, J. Algebra 144 (1991), p. 214-247. | Article | MR 1136904 | Zbl 0777.20019

[29] -, “Formes quadratiques de hauteur et de degré 2, Indag. Math. 7 (1996), p. 47-66. | Article | MR 1621344 | Zbl 0866.11030

[30] -, “La conjecture de Milnor, d'après V. Voevodsky”, in Séminaire Bourbaki (1996/97), Astérisque, vol. 245, Paris, 1997, Exp. 834, p. 379-418. | Numdam | Zbl 0916.19001

[31] B. Kahn, M. Rost & R. Sujatha - “Unramified cohomology of quadrics, I”, Amer. J. Math. 120 (1998), p. 841-891. | Article | MR 1637963 | Zbl 0913.11018

[32] N. Karpenko - “A relation between higher Witt indices”, Trans. Amer. Math. Soc., à paraître. | Article | MR 2279306 | Zbl 1148.11017

[33] -, “Invariants algébro-géométriques de formes quadratiques”, Algebra i Analiz 2 (1990), p. 141-162, en russe ; traduction anglaise : Leningrad Math. J. 2 (1991), p. 119-138.

[34] -, “Isotropie d'espaces cellulaires relatifs et de variétés de drapeaux isotropes”, Algebra i Analiz 12 (2000), p. 3-69, en russe ; traduction anglaise : St. Petersburg Math. J. 12 (2001), p. 1-50.

[35] -, “On the anisotropy of orthogonal involutions”, 15 (2000), p. 1-22. | MR 1751923 | Zbl 0962.16015

[36] -, “Characterization of minimal Pfister neighbors by Rost projectors”, J. Pure Appl. Algebra 160 (2001), p. 195-227. | Article | MR 1836000 | Zbl 0998.11016

[37] -, “On the first Witt index of quadratic forms”, Invent. Math. 153 (2003), p. 455-462. | Article | MR 1992018 | Zbl 1032.11016

[38] -, “Holes in I n , Ann. scient. Éc. Norm. Sup. 4 e série 37 (2004), p. 973-1002. | Numdam | MR 2119244 | Zbl 1108.11037

[39] N. Karpenko & A. Merkurjev - “Rost projectors and Steenrod operations”, 7 (2002), p. 481-493. | MR 2015051 | Zbl 1030.11013

[40] -, “Essential dimension of quadrics”, Invent. Math. 153 (2003), p. 361-372. | Article | MR 1992016 | Zbl 1032.11015

[41] I. Kersten & U. Rehmann - “Generic splitting of reductive groups”, 46 (1994), p. 35-70. | MR 1256727 | Zbl 0805.20034

[42] M. Knebusch - “Specialization of quadratic and symmetric bilinear forms, and a norm theorem”, Acta Arith. 24 (1973), p. 279-299. | Article | MR 349582 | Zbl 0287.15010

[43] -, “Generic splitting of quadratic forms, I”, Proc. London Math. Soc. (3) 33 (1976), p. 65-93. | MR 412101 | Zbl 0351.15016

[44] -, “Generic splitting of quadratic forms, II”, Proc. London Math. Soc. (3) 34 (1977), p. 1-31. | MR 427345 | Zbl 0359.15013

[45] T. Y. Lam - Introduction to quadratic forms over fields, Graduate Studies in Math., vol. 67, American Mathematical Society, Providence, 2005. | MR 2104929 | Zbl 1068.11023

[46] F. Loeser - “Cobordisme de variétés algébriques (d'après M. Levine et F. Morel)”, in Séminaire Bourbaki (2001/02), Astérisque, vol. 290, Paris, 2003, Exp. 901, p. 167-192. | Numdam | MR 2074055 | Zbl 1074.14020

[47] Yu. I. Manin - “Correspondances, motifs et transformations monoïdales”, Mat. Sb. 77 (119) (1968), p. 475-507, en russe ; traduction anglaise : Math. USSR Sbornik. | MR 258836 | Zbl 0199.24803

[48] F. Morel - “An introduction to 𝐀 1 -homotopy theory”, in Contemporary Developments in Algebraic K-theory (M. Karoubi, A. Kuku & C. Pedrini, éds.), ICTP Lect. Notes, vol. 15, 2003, p. 357-441. | MR 2175638 | Zbl 1081.14029

[49] F. Morel & V. Voevodsky - 𝐀 1 -homotopy theory of schemes”, Publ. Math. Inst. Hautes Études Sci. 90 (2001), p. 45-143. | Article | Numdam | MR 1813224 | Zbl 0983.14007

[50] D. Orlov, A. Vishik & V. Voevodsky - “An exact sequence for K * M /2 with applications to quadratic forms”, prépublication, www.math.uiuc.edu/K-theory, 2000. | Zbl 1124.14017

[51] E. Peyre - “Corps de fonctions de variétés homogènes et cohomologie galoisienne”, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), p. 891-896. | MR 1355848 | Zbl 0871.14038

[52] A. Pfister - “Multiplikative quadratische Formen”, Arch. Math. (Basel) 16 (1965), p. 363-370. | MR 184937 | Zbl 0146.26001

[53] M. Rost - “Some new results on the Chow groups of quadrics”, prépublication, Regensburg, 1990.

[54] L. Rowen - Ring theory, I, Pure and Applied Mathematics, vol. 127, Academic Press, 1988. | MR 940245 | Zbl 0651.16001

[55] W. Scharlau - “On the history of the algebraic theory of quadratic forms”, in Quadratic forms and their applications (Dublin, 1999), Contemp. Math., vol. 272, Providence, RI, 2000, p. 229-259. | MR 1803370 | Zbl 0974.11001

[56] J. E. Schneider - “Orthogonal groups of nonsingular forms of higher degree”, J. Algebra 27 (1973), p. 112-116. | Article | MR 325795 | Zbl 0287.20038

[57] A. Scholl - “Classical motives”, in Motives, Proc. Symp. Pure Math., vol. 55 (I), Providence, RI, 1994, p. 163-187. | MR 1265529 | Zbl 0814.14001

[58] T. A. Springer - “Sur les formes quadratiques d'indice zéro”, 234 (1952), p. 1517-1519. | MR 47021 | Zbl 0046.24303

[59] R. G. Swan - “Zero cycles on quadric hypersurfaces”, Proc. Amer. Math. Soc. 107 (1989), p. 43-46. | Article | MR 979219 | Zbl 0718.14007

[60] B. Totaro - “The Chow ring of a classifying space” 1997), Proc. Symp. Pure Math., vol. 67, Providence, RI, 1999, p. 249-281. | MR 1743244 | Zbl 0967.14005

[61] A. Vishik - “Integral motives of quadrics”, prépublication du Max Planck Institut für Mathematik (Bonn), 1998. | MR 2695910

[62] -, “On dimensions of anisotropic forms in I n , prépublication du Max Planck Institut für Mathematik (Bonn), 2000.

[63] -, “Motives of quadrics with applications to the theory of quadratic forms”, in Geometric methods in the algebraic theory of quadratic forms, Lect. Notes in Math., vol. 1835, Springer, 2004, p. 25-101. | MR 2066515 | Zbl 1047.11033

[64] V. Voevodsky - “Cancellation theorem”, prépublication, 2000. | Zbl 1202.14022

[65] -, “Triangulated categories of motives over a field”, in Cycles, transfers, and motivic homology theories, Ann. of Math. Stud., vol. 143, Princeton Univ. Press, Princeton, NJ, 2000, p. 188-238. | MR 1764202 | Zbl 1019.14009

[66] -, “Motivic cohomology with 𝐙/2-coefficients”, Publ. Math. Inst. Hautes Études Sci. 98 (2003), p. 59-104. | Article | Numdam | MR 2031199 | Zbl 1057.14028

[67] -, “Reduced power operations in motivic cohomology”, Publ. Math. Inst. Hautes Études Sci. 98 (2003), p. 1-57. | Article | Numdam | MR 2031198 | Zbl 1057.14027

[68] A. R. Wadsworth - “Noetherian pairs and function fields of quadratic forms”, Ph.D. Thesis, Université de Chicago, 1972. | MR 2611680

[69] -, “Similarity of quadratic forms and isomorphism of their function fields”, Trans. Amer. Math. Soc. 208 (1975), p. 352-358. | Article | MR 376527 | Zbl 0336.15013

[70] E. Witt - “Theorie der quadratischen Formen in beliebigen Körpern”, J. reine angew. Math. 176 (1937), p. 31-44. | JFM 62.0106.02 | MR 1581519