Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Composition algebras over algebraic curves of genus zero

Author: Holger P. Petersson
Journal: Trans. Amer. Math. Soc. 337 (1993), 473-493
MSC: Primary 17A75; Secondary 14H99, 17A45, 17D05
MathSciNet review: 1108613
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We rephrase the classical theory of composition algebras over fields, particularly the Cayley-Dickson Doubling Process and Zorn's Vector Matrices, in the setting of locally ringed spaces. Fixing an arbitrary base field, we use these constructions to classify composition algebras over (complete smooth) curves of genus zero. Applications are given to composition algebras over function fields of genus zero and polynomial rings.

References [Enhancements On Off] (What's this?)

  • [A] A. A. Albert, Quadratic forms permitting composition, Ann. of Math. (2) 43 (1942), 161–177. MR 0006140 (3,261a)
  • [BS] F. van der Blij and T. A. Springer, The arithmetics of octaves and of the group 𝐺₂, Nederl. Akad. Wetensch. Proc. Ser. A 62 = Indag. Math. 21 (1959), 406–418. MR 0152555 (27 #2533)
  • [EGA] A. Grothendieck, Éléments de géométrie algébrique. II, Inst. Hautes Études Sci. Publ. Math. 8 (1961).
  • [H] Günter Harder, Halbeinfache Gruppenschemata über vollständigen Kurven, Invent. Math. 6 (1968), 107–149 (German). MR 0263826 (41 #8425)
  • [Ha] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157 (57 #3116)
  • [K] M. Knebusch, Grothendieck- und Wittringe von nicht-ausgearteten symmetrischen Bilinearformen, Sitzungsber. Heidelb. Akad. Wiss. Math.-Natur. Kl., Springer-Verlag, Berlin, Heidelberg and New York, 1970.
  • [Kn] M.-A. Knus, Bundles on $ {{\mathbf{P}}^2}$ with a quaternionic structure. Proc. Conf. Vector Bundles on Algebraic Varieties, Tata Institute of Fundamental Research, Bombay, 1984.
  • [Kü] B. Küting, Kompositionsalgebren über rationalen Funktionenkörpern, Dissertation, FernUniversität, Hagen, 1987.
  • [KS] R. A. Kunze and S. Scheinberg, Alternative algebras having scalar involutions, Pacific J. Math. 124 (1986), no. 1, 159–172. MR 850673 (87i:17031)
  • [L] T. Y. Lam, Serre’s conjecture, Lecture Notes in Mathematics, Vol. 635, Springer-Verlag, Berlin, 1978. MR 0485842 (58 #5644)
  • [Le] Solange Legrand, Généralisations d’un théorème sur les algèbres munies d’une forme quadratique multiplicative, C. R. Acad. Sci. Paris Sér. A-B 274 (1972), A231–A234 (French). MR 0289595 (44 #6783)
  • [M] Kevin McCrimmon, Nonassociative algebras with scalar involution, Pacific J. Math. 116 (1985), no. 1, 85–109. MR 769825 (86d:17003)
  • [Mi] James S. Milne, Étale cohomology, Princeton Mathematical Series, vol. 33, Princeton University Press, Princeton, N.J., 1980. MR 559531 (81j:14002)
  • [PI] Holger P. Petersson, Composition algebras over a field with a discrete valuation, J. Algebra 29 (1974), 414–426. MR 0364380 (51 #635)
  • [P2] Holger P. Petersson, Composition algebras over the projective line, (Novosibirsk, 1989) Contemp. Math., vol. 131, Amer. Math. Soc., Providence, RI, 1992, pp. 673–678. MR 1175864
  • [S] Richard D. Schafer, An introduction to nonassociative algebras, Pure and Applied Mathematics, Vol. 22, Academic Press, New York, 1966. MR 0210757 (35 #1643)
  • [T] A. Tillmann, Unzerlegbare Vektorbündel über algebraischen Kurven, Dissertation, FernUniversität, Hagen, 1983.
  • [W] Ernst Witt, Über ein Gegenbeispiel zum Normensatz, Math. Z. 39 (1935), no. 1, 462–467 (German). MR 1545510,
  • [Z] L. Zagler, Ausgeartete, alternative, quadratische Algebren, Dissertation, Universität München, München, 1968.
  • [Zo] M. Zorn, Alternativkörper und quadratische Systeme, Abh. Math. Sem. Univ. Hamburg 9 (1933), 393-402.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 17A75, 14H99, 17A45, 17D05

Retrieve articles in all journals with MSC: 17A75, 14H99, 17A45, 17D05

Additional Information

PII: S 0002-9947(1993)1108613-X
Article copyright: © Copyright 1993 American Mathematical Society