Abstract
A field extension L / F is called excellent if, for any quadratic form φ over F, the anisotropic part (φL)an of φ over L is defined over F; L / F is called universally excellent if L ⋅ E / E is excellent for any field extension E / F. We study the excellence property for a generic splitting field of a central simple F-algebra. In particular, we show that it is universally excellent if and only if the Schur index of the algebra is not divisible by 4. We begin by studying the torsion in the second Chow group of products of Severi–Brauer varieties and its relationship with the relative Galois cohomology group H3(L / F) for a generic (common) splitting field L of the corresponding central simple F-algebras.
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References
Arason, J. Kr.: Cohomologische Invarianten quadratischer Formen, J. Algebra 36 (1975), 448-491.
Arason, J. Kr.: Excellence of F(φ)/F for 2-fold Pfister forms, Appendix II in [7] (1977).
Blanchet, A.: Function fields of generalized Brauer-Severi varieties, Comm. Algebra 19(1) (1991), 97-118.
Colliot-Thélène, J.-L. and, Sujatha, R.: Unramified Witt groups of real anisotropic quadrics, Proc. Sympos. Pure Math. 58(2) (1995), 127-147.
Elman, R.: Quadratic forms and the u-invariant, III, Queen's Papers Pure Appl. Math. 46 (1977), 422-444.
Elman, R. and Lam, T. Y.: Pfister forms and K-theory of fields, J. Algebra 23 (1972), 181-213.
Elman, R., Lam, T. Y. and Wadsworth, A. R.: Amenable fields and Pfister extensions, Queen's Papers Pure Appl. Math. 46 (1977), 445-491.
Esnault, H., Kahn, B., Levine, M. and Viehweg, V.: The Arason invariant and mod 2 algebraic cycles, J. Amer. Math. Soc. 11(1) (1998), 73-118.
Fulton, W.: Intersection Theory, Springer, New York, 1984.
Fulton, W. and Lang, S.: Riemann-Roch Algebra, Springer, New York, 1985.
Van Geel, J.: Applications of the Riemann-Roch theorem for curves to quadratic forms and division algebras, Preprint, Université Catholique de Louvain, 1991.
Hartshorne, R.: Algebraic Geometry, Springer, New York, 1977.
Hoffmann, D.W.: Splitting patterns and invariants of quadratic forms, Math. Nachr. 190 (1998), 149-168.
Hoffmann, D. W.: Twisted Pfister forms, Doc. Math. 1 (1996), 67-102.
Hurrelbrink, J. and Rehmann, U.: Splitting patterns of quadratic forms, Math. Nachr. 176 (1995), 111-127.
Izhboldin, O. T.: On the nonexcellence of field extensions F(π)/F, Doc. Math. 1 (1996), 127-136.
Izhboldin, O. T. and Karpenko, N. A.: Isotropy of 6-dimensional quadratic forms over function fields of quadrics, J. Algebra 209 (1998), 65-93.
Izhboldin, O. T. and Karpenko, N. A.: On the group H 3(F(ψ, D)/F), Doc. Math. 2 (1997), 297-311.
Karpenko, N. A.: Algebro-geometric invariants of quadratic forms, Algebra i Analiz 2(1) (1991), 141-162 (in Russian), Engl. transl.: Leningrad (St. Petersburg) Math. J. 2(1) (1991), 119–138.
Karpenko, N. A.: On topological filtration for Severi-Brauer varieties, Proc. Sympos. Pure Math. 58(2) (1995), 275-277.
Karpenko, N. A.: On topological filtration for Severi-Brauer varieties II, Amer. Math. Soc. Transl. 174(2) (1996), 45-48.
Karpenko, N. A.: Codimension 2 cycles on Severi-Brauer varieties, K-Theory 13(4) (1998), 305-330.
Karpenko, N. A. and Merkurjev, A. S.: Chow groups of projective quadrics, Algebra i Analiz 2(3) (1990), 218-235 (in Russian), Engl. transl.: Leningrad (St. Petersburg) Math. J. 2(3) (1991), 655–671.
Knebusch, M.: Generic splitting of quadratic forms, I, Proc. London Math. Soc. 33 (1976), 65-93.
Knebusch, M.: Generic splitting of quadratic forms, II, Proc. London Math. Soc. 34, (1977), 1-31.
Köck, B.: Chow motif and higher Chow theory of G/P, Manuscripta Math. 70 (1991), 363-372.
Lam, T. Y.: The Algebraic Theory of Quadratic Forms, Benjamin, Mass., 1973 (revised printing 1980).
Laghribi, A.: Formes quadratiques en 8 variables dont l'algèbre de Clifford est d'indice 8, K-Theory 12(4) (1997), 371-383.
Lewis, D. W. and Van Geel, J.: Quadratic forms isotropic over the function field of a conic, Indag. Math. 5 (1994), 325-339.
Mammone, P.: On the tensor product of division algebras, Arch. Math. 58 (1992), 34-39.
Manin, Yu. I.: Lectures on the K-functor in algebraic geometry, Russian Math. Surveys 24(5) (1969), 1-89.
Merkurjev, A. S.: On the norm residue symbol of degree 2, Dokl. Akad. Nauk SSSR 261 (1981), 542-547 (in Russian), Engl. transl.: Soviet Math. Dokl. 24 (1981), 546–551.
Merkurjev, A. S.: Kaplansky conjecture in the theory of quadratic forms, Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 175 (1989), 75-89 (in Russian), Engl. transl.: J. Soviet Math. 57(6) (1991), 3489–3497.
Merkurjev, A. S.: Simple algebras and quadratic forms, Izv. Akad. Nauk SSSR Ser. Mat. 55 (1991), 218-224 (in Russian), Engl. transl.: Math. USSR Izv. 38(1) (1992), 215–221.
Merkurjev, A. S. and Suslin, A. A.: The group K 3 for a field, Izv. Akad. Nauk SSSR Ser. Mat. 54(3) (1990), 522-545 (in Russian), Engl. transl.: Math. USSR, Izv. 36(3) (1991), 541–565.
Peyre, E.: Products of Severi-Brauer varieties and Galois cohomology, Proc. Sympos. Pure Math. 58(2) (1995), 369-401.
Pfister, A.: Quadratische Formen in beliebigen Körpern, Invent. Math. 1 (1966), 116-132.
Quillen, D.: Higher algebraic K-theory: I, In: H. Bass (ed.), Algebraic K-Theory I, Lecture Notes in Math. 341, Springer, New York, 1973, pp. 85-147.
Rost, M.: Hilbert 90 for K 3 for degree-two extensions, Preprint, 1986.
Rost, M.: Quadratic forms isotropic over the function field of a conic, Math. Ann. 288 (1990), 511-513.
Rowen, L. H.: Ring Theory, Volume II, Pure Appl. Math. 128, Academic Press, San Diego, 1988.
Sansuc, J.-J.: Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres, J. Reine Angew. Math. 327 (1981), 12-80.
Scharlau, W.: Quadratic and Hermitian Forms, Springer, New York, 1985.
Suslin, A. A.: Algebraic K-theory and norm-residue homomorphism, J. Soviet Math. 30 (1985), 2556-2611.
Schofield, A. and Van den Bergh, M.: The index of a Brauer class on a Brauer-Severi variety, Trans. Amer. Math. Soc. 333(2) (1992), 729-739.
Voevodsky, V.: The Milnor conjecture, Max-Planck-Institut für Mathematik in Bonn, Preprint MPI 97-8, 1997.
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Izhboldin, O.T., Karpenko, N.A. Generic Splitting Fields of Central Simple Algebras: Galois Cohomology and Nonexcellence. Algebras and Representation Theory 2, 19–59 (1999). https://doi.org/10.1023/A:1009910324736
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DOI: https://doi.org/10.1023/A:1009910324736