Quadratic forms and Pfister neighbors in characteristic 2
HTML articles powered by AMS MathViewer
- by Detlev W. Hoffmann and Ahmed Laghribi PDF
- Trans. Amer. Math. Soc. 356 (2004), 4019-4053 Request permission
Abstract:
We study Pfister neighbors and their characterization over fields of characteristic $2$, where we include the case of singular forms. We give a somewhat simplified proof of a theorem of Fitzgerald which provides a criterion for when a nonsingular quadratic form $q$ is similar to a Pfister form in terms of the hyperbolicity of this form over the function field of a form $\varphi$ which is dominated by $q$. From this, we derive an analogue in characteristic $2$ of a result by Knebusch saying that, in characteristic $\neq 2$, a form is a Pfister neighbor if its anisotropic part over its own function field is defined over the base field. Our result includes certain cases of singular forms, but we also give examples which show that Knebusch’s result generally fails in characteristic $2$ for singular forms. As an application, we characterize certain forms of height $1$ in the sense of Knebusch whose quasi-linear parts are of small dimension. We also develop some of the basics of a theory of totally singular quadratic forms. This is used to give a new interpretation of the notion of the height of a standard splitting tower as introduced by the second author in an earlier paper.References
- Hamza Ahmad, The algebraic closure in function fields of quadratic forms in characteristic $2$, Bull. Austral. Math. Soc. 55 (1997), no. 2, 293–297. MR 1438847, DOI 10.1017/S0004972700033955
- T. Venkatarayudu, The $7$-$15$ problem, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 531. MR 0000001, DOI 10.1090/gsm/058
- 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
- Jón Kr. Arason and Richard Elman, Powers of the fundamental ideal in the Witt ring, J. Algebra 239 (2001), no. 1, 150–160. MR 1827878, DOI 10.1006/jabr.2000.8688
- Jón Kristinn Arason and Albrecht Pfister, Beweis des Krullschen Durchschnittsatzes für den Wittring, Invent. Math. 12 (1971), 173–176 (German). MR 294251, DOI 10.1007/BF01404657
- Roberto Aravire and Ricardo Baeza, A note on generic splitting of quadratic forms, Comm. Algebra 27 (1999), no. 7, 3473–3477. MR 1695558, DOI 10.1080/00927879908826638
- R. Aravire, R. Baeza, The behaviour of quadratic and differential forms under function field extensions in characteristic $2$, J. Algebra 259 (2003), no. 2, 361–414.
- Roberto Aravire, Bill Jacob, and Pasquale Mammone, On the $u$-invariant for quadratic forms and the linkage of cyclic algebras, Math. Z. 214 (1993), no. 1, 137–146. MR 1234603, DOI 10.1007/BF02572396
- J. J. Corliss, Upper limits to the real roots of a real algebraic equation, Amer. Math. Monthly 46 (1939), 334–338. MR 4, DOI 10.1080/00029890.1939.11998880
- Ricardo Baeza, Ein Teilformensatz für quadratische Formen in Charakteristik $2$, Math. Z. 135 (1973/74), 175–184 (German). MR 337765, DOI 10.1007/BF01189354
- Ricardo Baeza, Quadratic forms over semilocal rings, Lecture Notes in Mathematics, Vol. 655, Springer-Verlag, Berlin-New York, 1978. MR 0491773, DOI 10.1007/BFb0070341
- Ricardo Baeza, The norm theorem for quadratic forms over a field of characteristic $2$, Comm. Algebra 18 (1990), no. 5, 1337–1348. MR 1059733, DOI 10.1080/00927879008823968
- Jean-Louis Colliot-Thélène and Alexei N. Skorobogatov, Groupe de Chow des zéro-cycles sur les fibrés en quadriques, $K$-Theory 7 (1993), no. 5, 477–500 (French, with English summary). MR 1255062, DOI 10.1007/BF00961538
- P. K. Draxl, Skew fields, London Mathematical Society Lecture Note Series, vol. 81, Cambridge University Press, Cambridge, 1983. MR 696937, DOI 10.1017/CBO9780511661907
- Robert W. Fitzgerald, Function fields of quadratic forms, Math. Z. 178 (1981), no. 1, 63–76. MR 627094, DOI 10.1007/BF01218371
- Detlev W. Hoffmann, Isotropy of quadratic forms over the function field of a quadric, Math. Z. 220 (1995), no. 3, 461–476. MR 1362256, DOI 10.1007/BF02572626
- Detlev W. Hoffmann, Splitting patterns and invariants of quadratic forms, Math. Nachr. 190 (1998), 149–168. MR 1611608, DOI 10.1002/mana.19981900108
- D. Hoffmann, A. Laghribi, Isotropy of quadratic forms over the function field of a quadric in characteristic $2$, Preprint 2002.
- Jürgen Hurrelbrink and Ulf Rehmann, Splitting patterns of excellent quadratic forms, J. Reine Angew. Math. 444 (1993), 183–192. MR 1241799
- Jürgen Hurrelbrink and Ulf Rehmann, Splitting patterns of quadratic forms, Math. Nachr. 176 (1995), 111–127. MR 1361129, DOI 10.1002/mana.19951760109
- Bruno Kahn, Formes quadratiques de hauteur et de degré 2, Indag. Math. (N.S.) 7 (1996), no. 1, 47–66 (French, with English summary). MR 1621344, DOI 10.1016/0019-3577(96)88656-3
- Manfred Knebusch, Specialization of quadratic and symmetric bilinear forms, and a norm theorem, Acta Arith. 24 (1973), 279–299. MR 349582, DOI 10.4064/aa-24-3-279-299
- Manfred Knebusch, Generic splitting of quadratic forms. I, Proc. London Math. Soc. (3) 33 (1976), no. 1, 65–93. MR 412101, DOI 10.1112/plms/s3-33.1.65
- Manfred Knebusch, Generic splitting of quadratic forms. II, Proc. London Math. Soc. (3) 34 (1977), no. 1, 1–31. MR 427345, DOI 10.1112/plms/s3-34.1.1
- M. Knebusch, Spezialisierung von quadratischen und symmetrisch bilinearen Formen, book in preparation.
- Manfred Knebusch and Ulf Rehmann, Generic splitting towers and generic splitting preparation of quadratic forms, Quadratic forms and their applications (Dublin, 1999) Contemp. Math., vol. 272, Amer. Math. Soc., Providence, RI, 2000, pp. 173–199. MR 1803367, DOI 10.1090/conm/272/04403
- Manfred Knebusch and Winfried Scharlau, Algebraic theory of quadratic forms, Birkhäuser, Boston, Mass., 1980. Generic methods and Pfister forms; Notes taken by Heisook Lee; DMV Seminar, 1. MR 583195
- Ahmed Laghribi, Certaines formes quadratiques de dimension au plus 6 et corps des fonctions en caractéristique 2, Israel J. Math. 129 (2002), 317–361 (French, with English summary). MR 1910948, DOI 10.1007/BF02773169
- Ahmed Laghribi, On the generic splitting of quadratic forms in characteristic 2, Math. Z. 240 (2002), no. 4, 711–730. MR 1922726, DOI 10.1007/s002090200387
- A. Laghribi, On splitting of totally singular quadratic forms, preprint (2003).
- A. Laghribi, P. Mammone, Isotropie d’une forme quadratique sur le corps des fonctions d’une quadrique projective en caractéristique $2$, Bull. Belg. Math. Soc. 9 (2002), 167–176.
- P. Mammone, J.-P. Tignol, and A. Wadsworth, Fields of characteristic $2$ with prescribed $u$-invariants, Math. Ann. 290 (1991), no. 1, 109–128. MR 1107665, DOI 10.1007/BF01459240
- A. S. Merkur′ev, Simple algebras and quadratic forms, Izv. Akad. Nauk SSSR Ser. Mat. 55 (1991), no. 1, 218–224 (Russian); English transl., Math. USSR-Izv. 38 (1992), no. 1, 215–221. MR 1130036
- D. Orlov, A. Vishik, V. Voevodsky, Motivic cohomology of Pfister quadrics and Milnor’s conjecture on quadratic forms, preprint (1998).
- Albrecht Pfister, Quadratic forms with applications to algebraic geometry and topology, London Mathematical Society Lecture Note Series, vol. 217, Cambridge University Press, Cambridge, 1995. MR 1366652, DOI 10.1017/CBO9780511526077
- Chih Han Sah, Symmetric bilinear forms and quadratic forms, J. Algebra 20 (1972), 144–160. MR 294378, DOI 10.1016/0021-8693(72)90094-4
- 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
- Adrian R. Wadsworth, Similarity of quadratic forms and isomorphism of their function fields, Trans. Amer. Math. Soc. 208 (1975), 352–358. MR 376527, DOI 10.1090/S0002-9947-1975-0376527-8
Additional Information
- Detlev W. Hoffmann
- Affiliation: Laboratoire de Mathématiques, UMR 6623 du CNRS, Université de Franche-Comté, 16 Route de Gray, F-25030 Besançon Cedex, France
- Address at time of publication: School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, Great Britain
- Email: detlev@math.univ-fcomte.fr, detlev.hoffmann@nottingham.ac.uk
- Ahmed Laghribi
- Affiliation: Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany
- Email: laghribi@mathematik.uni-bielefeld.de
- Received by editor(s): January 21, 2003
- Received by editor(s) in revised form: June 27, 2003
- Published electronically: February 27, 2004
- Additional Notes: Both authors have been supported in part by the European research networks FMRX-CT97-0107 and HPRN-CT-2002-00287 “Algebraic $K$-Theory, Linear Algebraic Groups and Related Structures”, and by the program INTAS 99-00817 “Linear Algebraic Groups and Related Linear and Homological Structures”.
- © Copyright 2004 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 356 (2004), 4019-4053
- MSC (2000): Primary 11E04; Secondary 11E81
- DOI: https://doi.org/10.1090/S0002-9947-04-03461-0
- MathSciNet review: 2058517