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The Arason invariant and mod 2 algebraic cycles


Authors: Hélène Esnault, Bruno Kahn, Marc Levine and Eckart Viehweg
Journal: J. Amer. Math. Soc. 11 (1998), 73-118
MSC (1991): Primary 11E81; Secondary 55R40
DOI: https://doi.org/10.1090/S0894-0347-98-00248-3
MathSciNet review: 1460391
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Abstract: Let $k$ be a field, $X$ over $k$ a smooth variety with function field $K$ and $E$ a quadratic vector bundle over $X$. Assuming that the generic fibre $q$ of $E$ is in $I^3K\subset W(K)$, we compute the image of its Arason invariant

\begin{displaymath}e^3(q)\in H^0(X,{\mathcal H}_{\mathrm{\acute{e}t}}^3({\mathbb Z}/2))\end{displaymath}

in $CH^2(X)/2$ by the $d_2$ differential of the Bloch-Ogus spectral sequence. This gives an obstruction to $e^3(q)$ being a global cohomology class.


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  • 1. J. Kr. Arason, Cohomologische Invarianten quadratischer Formen, J. Alg. 36 (1975), 448-491. MR 52:10592
  • 2. J. Barge, Une definition cohomologique de l'invariant d'Arason, preprint, 1995.
  • 3. S. Bloch and A. Ogus, Gersten's conjecture and the homology of schemes, Ann. Sci. Ecole Norm. Sup. 7 (1974), 181-202. MR 54:318
  • 4. S. Bloch and K. Kato, $p$-adic étale cohomology, Publ. Math. IHES 63 (1986), 107-152. MR 87k:14018
  • 5. R. Bott On torsion in Lie groups, Proc. Acad. Sci. USA 40 (1954), 586-588. MR 16:12a
  • 6. N. Bourbaki, Eléments de Mathématiques, Groupes et Algèbres de Lie, ch. 4,5,6, Masson, Paris, 1981. MR 83g:17001
  • 7. C. Chevalley, Sur les décompositions cellulaires des espaces $G/B$, Proc. Sympos. Pure Math. 56 (I), AMS, Providence, 1994, 1-23. MR 95e:14041
  • 8. J.-L. Colliot-Thélène, Birational invariants, purity and Gersten's conjecture, Proc. Sympos. Pure Math. 58.1, AMS, Providence, 1995, 1-64. MR 96c:14016
  • 9. J.-L. Colliot-Thélène and W. Raskind, $sK_2$-cohomology and the second Chow group, Math. Ann. 270 (1985), 165-199. MR 86m:14005
  • 10. J.-L. Colliot-Thélène, R. Hoobler, and B. Kahn, The Bloch-Ogus-Gabber theorem, to appear in: Proc. Fields Institute (Volume in memory of R. Thomason).
  • 11. P. Deligne, Théorie de Hodge, III, Publ. Math. IHES 44 (1974), 5-78. MR 58:16653b
  • 12. P. Deligne, unpublished notes of IHES lectures, 1979.
  • 13. M. Demazure and A. Grothendieck, Séminaire de géométrie algébrique du Bois-Marie: Schémas en groupes (SGA 3), tome III, Lecture notes in Math. 153, Springer, Berlin, 1970. MR 43:223c
  • 14. M. Demazure, Invariants symétriques du groupe de Weyl et torsion, Invent. Math. 21 (1973), 287-301. MR 49:7268
  • 15. M. Demazure, Désingularisation des variétés de Schubert généralisées, Ann. Sci. ENS 7 (1974), 53-68. MR 50:7174
  • 16. E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Mat. Sbornik N.S. 30(72) (1952), 349-462. Engl. translation: AMS Transl. Ser. II 6 (1957), 111-244. MR 13:904c
  • 17. H. Esnault, B. Kahn, and E. Viehweg, Coverings with odd ramification and Stiefel-Whitney classes, J. reine angew. Math. 441 (1993), 145-188. MR 94m:14017
  • 18. H. Freudenthal, Zur Berechnung der Charaktere der halbeinfachen Lieschen Gruppen. II, Indag. Math. 16 (1954), 487-491. MR 1:673a
  • 19. W. Fulton, Intersection theory, Erg. Math. 2, Springer, 1984. MR 85k:14004
  • 20. O. Gabber, Gersten's conjecture for some complexes of vanishing cycles, Manuscripta Math. 85 (1994), 323-343. MR 96m:14010
  • 21. H. Gillet, Riemann-Roch theorems for higher algebraic $K$-theory, Adv. in Math. 40 (1981), 203-289. MR 83m:14013
  • 22. M. Gros, Classes de Chern et classes de cycles en cohomologie de Hodge-Witt logarithmique, Mém. Soc. Math. France 21 (1985). MR 87m:14021
  • 23. A. Grothendieck, Torsion homologique et sections rationnelles, exposé 5 in Séminaire Chevalley, ``Anneaux de Chow et applications", Paris, 1958.
  • 24. A. Grothendieck, Problèmes ouverts en théorie des intersections, exposé XIV in Théorie des intersections et théorème de Riemann-Roch (SGA6), Lect. Notes in Math. 225, Springer, 1971, 667-689. MR 50:7133
  • 25. B. Harris, Torsion in Lie goups and related spaces, Topology 5 (1966), 347-354. MR 34:6798
  • 26. J.E. Humphreys, Linear algebraic groups (corrected third printing), Springer, New York, 1987. MR 53:633 (original 1975 printing)
  • 27. L. Illusie, Complexe cotangent et déformations I, Lect. Notes in Math. 239, Springer, Berlin, 1971. MR 58:10886a
  • 28. W. Jacob and M. Rost, Degree four cohomological invariants for quadratic forms, Invent. Math. 96 (1989), 551-570. MR 90g:11044
  • 29. U. Jannsen, Mixed Motives and Algebraic $K$-Theory, Lecture Notes in Mathematics 1400, Springer, Berlin-Heidelberg, 1990. MR 91g:14008
  • 30. J.F. Jardine, Higher spinor classes, Mem. Amer. Math. Soc. 110 (1994), no. 528. MR 95a:11035
  • 31. J.-P. Jouanolou, Riemann-Roch sans dénominateurs, Invent. Math. 11 (1970), 15-26. MR 48:11115
  • 32. B. Kahn, Descente galoisienne et $K_2$ des corps de nombres, $K$-theory 7 (1993), 55-100. MR 94i:11094
  • 33. B. Kahn, Applications of weight-two motivic cohomology, Documenta Math. 1 (1996), 395-416. CMP 97:05
  • 34. A. Laghribi, Isotropie de certaines formes quadratiques de dimension 7 et 8 sur le corps des fonctions d'une quadrique, Duke Math. J. 85 (1996), 397-410. MR 97h:11039
  • 35. T.Y. Lam, The algebraic theory of quadratic forms (2nd ed.), Benjamin, New York, 1980. MR 83d:10022
  • 36. Y. Laszlo and C. Sorger, The line bundles on the moduli of parabolic $G$-bundles over curves and their sections, Ann. Sci. Ec. Norm. Sup. (4) 30 (1997), no. 4, 499-525. CMP 97:14
  • 37. M. Levine, The indecomposable $K_3$ of fields, Ann. Sci. Ec. Norm. Sup. 22 (1989), 255-344. MR 91a:11061
  • 38. M. Levine, The algebraic $K$-theory of the classical groups and some twisted forms, Duke Math. J. 70 (1993), 405-443. MR 94d:19004
  • 39. S. Lichtenbaum, Values of zeta-functions at non-negative integers, Lect. Notes in Math. 1068, Springer, Berlin, 1984, 127-138. CMP 16:17
  • 40. S. Lichtenbaum, The construction of weight-two arithmetic cohomology, Invent. Math. 88 (1987), 183-215. MR 88d:14011
  • 41. S. Lichtenbaum, New results on weight-two motivic cohomology, The Grothendieck Festschrift, vol. 3, Progress in Math. 88, Birkhaüser, Boston, 1990, 35-55. MR 92m:14030
  • 42. R. Marlin, Anneaux de Chow des groupes algébriques $ \text{ SU}(n)$, $ \text{Sp}(n)$, $ \text{SO}(n)$, $ \text{Spin}(n)$, $G\sb{2}$, $F\sb{4}$; torsion, C. R. Acad. Sci. Paris 279 (1974), 119-122. MR 50:321
  • 43. A. S. Merkurjev, On the norm residue symbol of degree $2$, Dokl. Akad. Nauk SSSR 261 (1981), 542-547. English translation: Soviet Math. Dokl. 24 (1981), 546-551. MR 83h:12015
  • 44. A. S. Merkurjev, The group $H^1(X,{\mathcal K}_2)$ for projective homogeneous varieties (in Russian), Algebra i Analiz 7 (1995), 136-164. English translation: Leningrad (Saint-Petersburg) Math. J. 7 (1996), 421-444. MR 97a:19003
  • 45. A. S. Merkurjev, Comparison of equivariant and ordinary $K$-theory of algebraic varieties, to appear in St.-Petersburg Math. J.
  • 46. A. S. Merkurjev and A. A. Suslin, $\cal K$-cohomology of Severi-Brauer varieties and norm residue homomorphism (in Russian), Izv. Akad. Nauk SSSR 46 (1982), 1011-1046. English translation: Math USSR Izv. 21 (1983), 307-340. MR 84i:12007
  • 47. A. S. Merkurjev and A. A. Suslin, The norm residue homomorphism of degree $3$ (in Russian), Izv. Akad. Nauk SSSR 54 (1990), 339-356. English translation: Math. USSR Izv. 36 (1991), 349-368. MR 91f:11083
  • 48. A. S. Merkurjev and A. A. Suslin, The group $K_3$ for a field (in Russian), Izv. Akad. Nauk. SSSR 54 (1990), 522-545. English translation: Math. USSR Izv. 36 (1991), 541-565. MR 91g:19002
  • 49. J. W. Milnor and J. D. Stasheff, Characteristic classes, Annals of Mathematics Studies 76, Princeton University Press, Princeton, 1974. MR 55:13428
  • 50. I. A. Panin, Application of $K$-theory in algebraic geometry, doctoral dissertation, LOMI, Leningrad, 1984.
  • 51. I. A. Panin, A splitting principle, Preprint, Bielefeld University, 1994.
  • 52. R. Parimala and V. Srinivas, Analogues of the Brauer group for algebras with involution, Duke Math. J. 66 (1992), 207-237. MR 93i:16027
  • 53. E. Peyre, Corps de fonctions de variétés homogènes et cohomologie galoisienne, C. R. Acad. Sci. Paris 321 (1995), 891-896. MR 96i:14006
  • 54. D. Quillen, Higher algebraic $K$-theory, I, Lect. Notes in Math. 341, Springer, New York, 1973, 83-147. MR 49:2895
  • 55. P. Roquette, The Galois cohomology of the projective linear group, Math. Ann. 150 (1963), 411-439. MR 27:4832
  • 56. M. Rost, Hilbert's theorem 90 for $K_3^M$ for degree-two extensions, preprint, Regensburg, 1986.
  • 57. M. Rost, Chow groups with coefficients, Documenta Math. 1 (1996), 319-393. CMP 97:04
  • 58. M. Rost, Cohomological invariants, in preparation.
  • 59. W. Scharlau, Quadratic and hermitian forms, Springer, Berlin, 1985. MR 86k:11022
  • 60. J-P. Serre, Cohomologie galoisienne: progrès et problèmes, Sém. N. Bourbaki, march 1994, exposé 783. MR 97d:11063
  • 61. C. S. Seshadri, Line bundles on Schubert varieties, Vector bundles on algebraic varieties (Bombay, 1984), T.I.F.R. Studies in Math. 11, Oxford University Press, 1987, 499-528. MR 88i:14047
  • 62. A. A. Suslin, $K$-theory and ${\mathcal K}$-cohomology of certain group varieties, Adv. in Soviet Math. 4, AMS, Providence, 1991, 53-74. MR 92g:19004
  • 63. A. A. Suslin, Torsion in $K_2$ of fields, $K$-theory 1 (1987), 5-29. MR 89a:11123
  • 64. M. Szyjewski, The fifth invariant of quadratic forms (in Russian), Algebra Anal. 2 (1990), 213-234. English translation: Leningrad Math. J. 2 (1991), 179-198. MR 91d:11040
  • 65. J. Tits, Classification of algebraic semi-simple groups, Proc. Sympos. Pure Math. 9, AMS, 1966, 33-62. MR 37:309

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Additional Information

Hélène Esnault
Affiliation: FB6, Mathematik, Universität Essen, D-45117 Essen, Germany
Email: esnault@uni-essen.de

Bruno Kahn
Affiliation: Institut de Mathématiques de Jussieu, Université Paris 7, Case 7012, 75251 Paris Cedex 05, France
Email: kahn@math.jussieu.fr

Marc Levine
Affiliation: Department of Mathematics, Northeastern University, Boston, Massachusetts 02115
Email: marc@neu.edu

Eckart Viehweg
Affiliation: FB6, Mathematik, Universität Essen, D-45117 Essen, Germany
Email: viehweg@uni-essen.de

DOI: https://doi.org/10.1090/S0894-0347-98-00248-3
Received by editor(s): September 13, 1996
Received by editor(s) in revised form: July 28, 1997
Additional Notes: This research was partially supported by the DFG Forschergruppe “Arithmetik und Geometrie”; the second and third author gratefully acknowledge its hospitality.
Article copyright: © Copyright 1998 American Mathematical Society

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