On standard norm varieties

Karpenko, Nikita A.; Merkurjev, Alexander S.

doi:10.24033/asens.2187

On standard norm varieties
[Sur les variétés de norme standard]

Karpenko, Nikita A. ; Merkurjev, Alexander S.

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 46 (2013) no. 1, pp. 177-216.

Résumé
Abstract

Pour un nombre premier $p$ et un corps $F$ de caractéristique $0$ , soit $X$ la variété de norme d’un symbole dans le groupe de cohomologie galoisienne $H^{n + 1} (F, μ_{p}^{\otimes n})$ (avec $n \geq 1$ ) construite au cours de la démonstration de la conjecture de Bloch-Kato. Le résultat principal de cet article affirme que le corps des fonctions $F (X)$ a la propriété suivante : pour toute variété équidimensionnelle $Y$ , l’homomorphisme de changement de corps $CH (Y) \to CH (Y_{F (X)})$ de groupes de Chow à coefficients entiers localisés en $p$ est surjectif en codimension $< (dim X) / (p - 1)$ . Une des composantes principales de la preuve est le calcul de groupes de Chow du motif de Rost généralisé (un variant du résultat principal indépendant de ceci est proposé dans l’appendice). Un autre ingrédient important est la $A$ -trivialité de $X$ , la propriété qui dit que pour toute extension de corps $L / F$ avec $X (L) \neq \emptyset$ , l’homomorphisme de degré pour ${CH}_{0} (X_{L})$ est injectif. La preuve fait apparaître la théorie de correspondances rationnelles revue dans l’appendice.

Let $p$ be a prime integer and $F$ a field of characteristic $0$ . Let $X$ be the norm variety of a symbol in the Galois cohomology group $H^{n + 1} (F, μ_{p}^{\otimes n})$ (for some $n \geq 1$ ), constructed in the proof of the Bloch-Kato conjecture. The main result of the paper affirms that the function field $F (X)$ has the following property: for any equidimensional variety $Y$ , the change of field homomorphism $CH (Y) \to CH (Y_{F (X)})$ of Chow groups with coefficients in integers localized at $p$ is surjective in codimensions $< (dim X) / (p - 1)$ . One of the main ingredients of the proof is a computation of Chow groups of a (generalized) Rost motive (a variant of the main result not relying on this is given in the appendix). Another important ingredient is $A$ -triviality of $X$ , the property saying that the degree homomorphism on ${CH}_{0} (X_{L})$ is injective for any field extension $L / F$ with $X (L) \neq \emptyset$ . The proof involves the theory of rational correspondences reviewed in the appendix.

DOI : 10.24033/asens.2187

Classification : 14C25
Keywords: norm varieties, Chow groups and motives, Steenrod operations
Mot clés : variétés de norme, groupes et motifs de Chow, opérations de Steenrod

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     author = {Karpenko, Nikita A. and Merkurjev, Alexander S.},
     title = {On standard norm varieties},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {177--216},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 46},
     number = {1},
     year = {2013},
     doi = {10.24033/asens.2187},
     language = {en},
     url = {https://www.numdam.org/articles/10.24033/asens.2187/}
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Karpenko, Nikita A.; Merkurjev, Alexander S. On standard norm varieties. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 46 (2013) no. 1, pp. 177-216. doi : 10.24033/asens.2187. https://www.numdam.org/articles/10.24033/asens.2187/

Bibliographie
Cité par

[1] A. R. Boisvert, A new definition of the Steenrod operations in algebraic geometry, ProQuest LLC, Ann Arbor, MI, 2007, Thesis (Ph.D.)-University of California, Los Angeles. | MR

[2] A. R. Boisvert, A new definition of the Steenrod operations in algebraic geometry, preprint arXiv:0805.1414.

[3] P. Brosnan, A short proof of Rost nilpotence via refined correspondences, Doc. Math. 8 (2003), 69-78. | MR

[4] P. Brosnan, Steenrod operations in Chow theory, Trans. Amer. Math. Soc. 355 (2003), 1869-1903. | MR

[5] V. Chernousov & A. S. Merkurjev, Motivic decomposition of projective homogeneous varieties and the Krull-Schmidt theorem, Transform. Groups 11 (2006), 371-386. | MR

[6] P. K. Draxl, Skew fields, London Mathematical Society Lecture Note Series 81, Cambridge Univ. Press, 1983. | MR

[7] R. Elman, N. Karpenko & A. Merkurjev, The algebraic and geometric theory of quadratic forms, American Mathematical Society Colloquium Publications 56, Amer. Math. Soc., 2008. | MR

[8] R. Fino, Around rationality of integral cycles, J. Pure Appl. Algebra (2012), doi://10.1016/j.jpaa.2012.12.003. | MR

[9] R. Fino, Around rationality of cycles, to appear in Cent. Eur. J. Math. | MR

[10] W. Fulton, Intersection theory, second éd., Ergeb. Math. Grenzg. 2, Springer, 1998. | MR

[11] S. Garibaldi, Cohomological invariants: exceptional groups and spin groups, Mem. Amer. Math. Soc. 200 (2009), 81. | MR

[12] A. Grothendieck, Techniques de construction et théorèmes d'existence en géométrie algébrique. IV. Les schémas de Hilbert, in Séminaire Bourbaki, vol. 6, exp. no 221, Soc. Math. France, 1995, 249-276. | Numdam | MR

[13] B. Kahn & R. Sujatha, Birational motives, I, $K$ -theory Preprint Archives (preprint server) 596, 2002.

[14] N. Karpenko & A. Merkurjev, Rost projectors and Steenrod operations, Doc. Math. 7 (2002), 481-493. | MR

[15] N. A. Karpenko, Criteria of motivic equivalence for quadratic forms and central simple algebras, Math. Ann. 317 (2000), 585-611. | MR

[16] N. A. Karpenko, Weil transfer of algebraic cycles, Indag. Math. (N.S.) 11 (2000), 73-86. | MR

[17] N. A. Karpenko, Upper motives of algebraic groups and incompressibility of Severi-Brauer varieties, J. reine angew. Math. (2012), doi://10.1515/crelle.2012.011. | MR

[18] N. A. Karpenko & A. S. Merkurjev, Canonical $p$ -dimension of algebraic groups, Adv. Math. 205 (2006), 410-433. | MR

[19] J. I. Manin, Correspondences, motifs and monoidal transformations, Mat. Sb. (N.S.) 77 (119) (1968), 475-507. | MR

[20] A. S. MerkurʼEv & A. Suslin, $K$ -cohomology of Severi-Brauer varieties and the norm residue homomorphism, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), 1011-1046, 1135-1136. | MR

[21] A. Merkurjev, Rost invariants of simply connected algebraic groups, in Cohomological invariants in Galois cohomology, Univ. Lecture Ser. 28, Amer. Math. Soc., 2003, 101-158. | MR

[22] A. Merkurjev, Unramified elements in cycle modules, J. Lond. Math. Soc. 78 (2008), 51-64. | MR

[23] A. Merkurjev & A. Suslin, Motivic cohomology of the simplicial motive of a Rost variety, J. Pure Appl. Algebra 214 (2010), 2017-2026. | MR

[24] A. S. Merkurjev, Essential dimension, in Quadratic forms-algebra, arithmetic, and geometry, Contemp. Math. 493, Amer. Math. Soc., 2009, 299-325. | MR

[25] D. H. Nguyen, On p-generic splitting varieties for Milnor K-symbols mod p, ProQuest LLC, Ann Arbor, MI, 2009, Thesis (Ph.D.)-University of California, Los Angeles. | MR

[26] D. H. Nguyen, On $p$ -generic splitting varieties for Milnor $K$ -symbols $\mod p$ , preprint arXiv:1003.3971. | MR

[27] I. Panin, Application of $K$ -theory in algebraic geometry, Thèse, LOMI, Leningrad, 1984.

[28] M. Rost, Chow groups with coefficients, Doc. Math. 1 (1996), No. 16, 319-393. | MR

[29] M. Rost, On the basic correspondence of a splitting variety, preprint http://www.math.uni-bielefeld.de/~rost/data/bkc-c.pdf.

[30] A. Suslin & S. Joukhovitski, Norm varieties, J. Pure Appl. Algebra 206 (2006), 245-276. | MR

[31] M. L. Thakur, Isotopy and invariants of Albert algebras, Comment. Math. Helv. 74 (1999), 297-305. | MR

[32] A. Vishik, Generic points of quadrics and Chow groups, Manuscripta Math. 122 (2007), 365-374. | MR

[33] A. Vishik, Rationality of integral cycles, Doc. Math. (2010), 661-670, Extra volume: Andrei A. Suslin sixtieth birthday. | MR

[34] A. Vishik & K. Zainoulline, Motivic splitting lemma, Doc. Math. 13 (2008), 81-96. | MR

[35] V. Voevodsky, On motivic cohomology with $𝐙 / l$ -coefficients, Ann. of Math. 174 (2011), 401-438. | MR

[36] K. Zainoulline, Special correspondences and Chow traces of Landweber-Novikov operations, J. reine angew. Math. 628 (2009), 195-204. | MR

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