The Arason invariant and mod 2 algebraic cycles
HTML articles powered by AMS MathViewer
- by Hélène Esnault, Bruno Kahn, Marc Levine and Eckart Viehweg
- J. Amer. Math. Soc. 11 (1998), 73-118
- DOI: https://doi.org/10.1090/S0894-0347-98-00248-3
- PDF | Request permission
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 \[ e^3(q)\in H^0(X,{\mathcal H}_{\mathrm {\acute {e}t}}^3({\mathbb Z}/2))\] 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.References
- Jón Kr. Arason, Cohomologische invarianten quadratischer Formen, J. Algebra 36 (1975), no. 3, 448–491 (French). MR 389761, DOI 10.1016/0021-8693(75)90145-3
- J. Barge, Une definition cohomologique de l’invariant d’Arason, preprint, 1995.
- Spencer Bloch and Arthur Ogus, Gersten’s conjecture and the homology of schemes, Ann. Sci. École Norm. Sup. (4) 7 (1974), 181–201 (1975). MR 412191, DOI 10.24033/asens.1266
- Spencer Bloch and Kazuya Kato, $p$-adic étale cohomology, Inst. Hautes Études Sci. Publ. Math. 63 (1986), 107–152. MR 849653, DOI 10.1007/BF02831624
- Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
- Nicolas Bourbaki, Éléments de mathématique, Masson, Paris, 1981 (French). Groupes et algèbres de Lie. Chapitres 4, 5 et 6. [Lie groups and Lie algebras. Chapters 4, 5 and 6]. MR 647314
- C. Chevalley, Sur les décompositions cellulaires des espaces $G/B$, Algebraic groups and their generalizations: classical methods (University Park, PA, 1991) Proc. Sympos. Pure Math., vol. 56, Amer. Math. Soc., Providence, RI, 1994, pp. 1–23 (French). With a foreword by Armand Borel. MR 1278698, DOI 10.1090/pspum/056.1/1278698
- J.-L. Colliot-Thélène, Birational invariants, purity and the Gersten conjecture, $K$-theory and algebraic geometry: connections with quadratic forms and division algebras (Santa Barbara, CA, 1992) Proc. Sympos. Pure Math., vol. 58, Amer. Math. Soc., Providence, RI, 1995, pp. 1–64. MR 1327280, DOI 10.1090/pspum/058.1/1327280
- Jean-Louis Colliot-Thélène and Wayne Raskind, ${\scr K}_2$-cohomology and the second Chow group, Math. Ann. 270 (1985), no. 2, 165–199. MR 771978, DOI 10.1007/BF01456181
- 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).
- Pierre Deligne, Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math. 40 (1971), 5–57 (French). MR 498551, DOI 10.1007/BF02684692
- P. Deligne, unpublished notes of IHES lectures, 1979.
- Schémas en groupes. I: Propriétés générales des schémas en groupes, Lecture Notes in Mathematics, Vol. 151, Springer-Verlag, Berlin-New York, 1970 (French). Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3); Dirigé par M. Demazure et A. Grothendieck. MR 0274458
- Michel Demazure, Invariants symétriques entiers des groupes de Weyl et torsion, Invent. Math. 21 (1973), 287–301 (French). MR 342522, DOI 10.1007/BF01418790
- Michel Demazure, Désingularisation des variétés de Schubert généralisées, Ann. Sci. École Norm. Sup. (4) 7 (1974), 53–88 (French). MR 354697, DOI 10.24033/asens.1261
- C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
- Hélène Esnault, Bruno Kahn, and Eckart Viehweg, Coverings with odd ramification and Stiefel-Whitney classes, J. Reine Angew. Math. 441 (1993), 145–188. MR 1228615
- T. Venkatarayudu, The $7$-$15$ problem, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 531. MR 0000001, DOI 10.1090/gsm/058
- William Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. MR 732620, DOI 10.1007/978-3-662-02421-8
- Ofer Gabber, Gersten’s conjecture for some complexes of vanishing cycles, Manuscripta Math. 85 (1994), no. 3-4, 323–343. MR 1305746, DOI 10.1007/BF02568202
- Henri Gillet, Riemann-Roch theorems for higher algebraic $K$-theory, Adv. in Math. 40 (1981), no. 3, 203–289. MR 624666, DOI 10.1016/S0001-8708(81)80006-0
- Michel Gros, Classes de Chern et classes de cycles en cohomologie de Hodge-Witt logarithmique, Mém. Soc. Math. France (N.S.) 21 (1985), 87 (French, with English summary). MR 844488
- A. Grothendieck, Torsion homologique et sections rationnelles, exposé 5 in Séminaire Chevalley, “Anneaux de Chow et applications", Paris, 1958.
- Théorie des intersections et théorème de Riemann-Roch, Lecture Notes in Mathematics, Vol. 225, Springer-Verlag, Berlin-New York, 1971 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6); Dirigé par P. Berthelot, A. Grothendieck et L. Illusie. Avec la collaboration de D. Ferrand, J. P. Jouanolou, O. Jussila, S. Kleiman, M. Raynaud et J. P. Serre. MR 0354655
- Bruno Harris, Torsion in Lie groups and related spaces, Topology 5 (1966), 347–354. MR 206982, DOI 10.1016/0040-9383(66)90026-7
- James E. Humphreys, Linear algebraic groups, Graduate Texts in Mathematics, No. 21, Springer-Verlag, New York-Heidelberg, 1975. MR 0396773, DOI 10.1007/978-1-4684-9443-3
- Luc Illusie, Complexe cotangent et déformations. I, Lecture Notes in Mathematics, Vol. 239, Springer-Verlag, Berlin-New York, 1971 (French). MR 0491680, DOI 10.1007/BFb0059052
- Bill Jacob and Markus Rost, Degree four cohomological invariants for quadratic forms, Invent. Math. 96 (1989), no. 3, 551–570. MR 996554, DOI 10.1007/BF01393696
- Uwe Jannsen, Mixed motives and algebraic $K$-theory, Lecture Notes in Mathematics, vol. 1400, Springer-Verlag, Berlin, 1990. With appendices by S. Bloch and C. Schoen. MR 1043451, DOI 10.1007/BFb0085080
- J. F. Jardine, Higher spinor classes, Mem. Amer. Math. Soc. 110 (1994), no. 528, vi+88. MR 1211372, DOI 10.1090/memo/0528
- J. P. Jouanolou, Riemann-Roch sans dénominateurs, Invent. Math. 11 (1970), 15–26. MR 332789, DOI 10.1007/BF01389802
- Bruno Kahn, Descente galoisienne et $K_2$ des corps de nombres, $K$-Theory 7 (1993), no. 1, 55–100 (French, with English and French summaries). MR 1220427, DOI 10.1007/BF00962794
- B. Kahn, Applications of weight-two motivic cohomology, Documenta Math. 1 (1996), 395–416.
- Ahmed Laghribi, Isotropie de certaines formes quadratiques de dimensions $7$ et $8$ sur le corps des fonctions d’une quadrique, Duke Math. J. 85 (1996), no. 2, 397–410 (French). MR 1417621, DOI 10.1215/S0012-7094-96-08516-6
- T. Y. Lam, The algebraic theory of quadratic forms, Mathematics Lecture Note Series, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1980. Revised second printing. MR 634798
- 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.
- Marc Levine, The indecomposable $K_3$ of fields, Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 2, 255–344. MR 1005161, DOI 10.24033/asens.1585
- Marc Levine, The algebraic $K$-theory of the classical groups and some twisted forms, Duke Math. J. 70 (1993), no. 2, 405–443. MR 1219818, DOI 10.1215/S0012-7094-93-07008-1
- S. Lichtenbaum, Values of zeta-functions at non-negative integers, Lect. Notes in Math. 1068, Springer, Berlin, 1984, 127–138.
- Stephen Lichtenbaum, The construction of weight-two arithmetic cohomology, Invent. Math. 88 (1987), no. 1, 183–215. MR 877012, DOI 10.1007/BF01405097
- S. Lichtenbaum, New results on weight-two motivic cohomology, The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, Birkhäuser Boston, Boston, MA, 1990, pp. 35–55. MR 1106910, DOI 10.1007/978-0-8176-4576-2_{2}
- Roger Marlin, Anneaux de Chow des groupes algébriques $\textrm {SU}(n)$, $\textrm {Sp}(n)$, $\textrm {SO}(n)$, $\textrm {Spin}(n)$, $G_{2}$, $F_{4}$; torsion, C. R. Acad. Sci. Paris Sér. A 279 (1974), 119–122 (French). MR 347820
- A. S. Merkur′ev, On the norm residue symbol of degree $2$, Dokl. Akad. Nauk SSSR 261 (1981), no. 3, 542–547 (Russian). MR 638926
- A. S. Merkur′ev, The group $H^1(X,\scr K_2)$ for projective homogeneous varieties, Algebra i Analiz 7 (1995), no. 3, 136–164 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 7 (1996), no. 3, 421–444. MR 1353493
- A. S. Merkurjev, Comparison of equivariant and ordinary $K$-theory of algebraic varieties, to appear in St.-Petersburg Math. J.
- A. S. Merkur′ev and A. A. Suslin, $K$-cohomology of Severi-Brauer varieties and the norm residue homomorphism, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 5, 1011–1046, 1135–1136 (Russian). MR 675529
- A. S. Merkur′ev and A. A. Suslin, Norm residue homomorphism of degree three, Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 2, 339–356 (Russian); English transl., Math. USSR-Izv. 36 (1991), no. 2, 349–367. MR 1062517
- A. S. Merkur′ev and A. A. Suslin, The group $K_3$ for a field, Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 3, 522–545 (Russian); English transl., Math. USSR-Izv. 36 (1991), no. 3, 541–565. MR 1072694
- John W. Milnor and James D. Stasheff, Characteristic classes, Annals of Mathematics Studies, No. 76, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. MR 0440554, DOI 10.1515/9781400881826
- I. A. Panin, Application of $K$-theory in algebraic geometry, doctoral dissertation, LOMI, Leningrad, 1984.
- I. A. Panin, A splitting principle, Preprint, Bielefeld University, 1994.
- R. Parimala and V. Srinivas, Analogues of the Brauer group for algebras with involution, Duke Math. J. 66 (1992), no. 2, 207–237. MR 1162189, DOI 10.1215/S0012-7094-92-06606-3
- Emmanuel Peyre, Corps de fonctions de variétés homogènes et cohomologie galoisienne, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), no. 7, 891–896 (French, with English and French summaries). MR 1355848
- Daniel Quillen, Higher algebraic $K$-theory. I, Algebraic $K$-theory, I: Higher $K$-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Lecture Notes in Math., Vol. 341, Springer, Berlin, 1973, pp. 85–147. MR 0338129
- Peter Roquette, On the Galois cohomology of the projective linear group and its applications to the construction of generic splitting fields of algebras, Math. Ann. 150 (1963), 411–439. MR 154888, DOI 10.1007/BF01357435
- M. Rost, Hilbert’s theorem 90 for $K_3^M$ for degree-two extensions, preprint, Regensburg, 1986.
- M. Rost, Chow groups with coefficients, Documenta Math. 1 (1996), 319–393.
- M. Rost, Cohomological invariants, in preparation.
- Winfried Scharlau, Quadratic and Hermitian forms, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 270, Springer-Verlag, Berlin, 1985. MR 770063, DOI 10.1007/978-3-642-69971-9
- Jean-Pierre Serre, Cohomologie galoisienne: progrès et problèmes, Astérisque 227 (1995), Exp. No. 783, 4, 229–257 (French, with French summary). Séminaire Bourbaki, Vol. 1993/94. MR 1321649
- C. S. Seshadri, Line bundles on Schubert varieties, Vector bundles on algebraic varieties (Bombay, 1984) Tata Inst. Fund. Res. Stud. Math., vol. 11, Tata Inst. Fund. Res., Bombay, 1987, pp. 499–528. MR 893610
- A. A. Suslin, $K$-theory and $K$-cohomology of certain group varieties, Algebraic $K$-theory, Adv. Soviet Math., vol. 4, Amer. Math. Soc., Providence, RI, 1991, pp. 53–74. MR 1124626
- A. A. Suslin, Torsion in $K_2$ of fields, $K$-Theory 1 (1987), no. 1, 5–29. MR 899915, DOI 10.1007/BF00533985
- M. Shyevski, The fifth invariant of quadratic forms, Algebra i Analiz 2 (1990), no. 1, 213–234 (Russian); English transl., Leningrad Math. J. 2 (1991), no. 1, 179–198. MR 1049911
- J. Tits, Classification of algebraic semisimple groups, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) Amer. Math. Soc., Providence, R.I., 1966, pp. 33–62. MR 0224710
Bibliographic Information
- Hélène Esnault
- Affiliation: FB6, Mathematik, Universität Essen, D-45117 Essen, Germany
- MR Author ID: 64210
- 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
- MR Author ID: 113315
- Email: marc@neu.edu
- Eckart Viehweg
- Affiliation: FB6, Mathematik, Universität Essen, D-45117 Essen, Germany
- Email: viehweg@uni-essen.de
- 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.
- © Copyright 1998 American Mathematical Society
- 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