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Composition algebras over rings of genus zero


Author: S. Pumplün
Journal: Trans. Amer. Math. Soc. 351 (1999), 1277-1292
MSC (1991): Primary 17A75; Secondary 14H99
DOI: https://doi.org/10.1090/S0002-9947-99-01935-2
MathSciNet review: 1422907
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Abstract: The theory of composition algebras over locally ringed spaces and some basic results from algebraic geometry are used to characterize composition algebras over open dense subschemes of curves of genus zero.


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

  • [A] A. A. Albert, Quadratic forms permitting composition, Ann. Math. (2) 43 (1942), 161-177. MR 3:261a
  • [BJ] S. Balcerzyk and T. Jo\'{z}efiak, Commutative Noetherian and Krull Rings, Ellis Horwood, Chichester, 1989. MR 92f:13001
  • [BS] F. van der Blij and T. A. Springer, The arithmetics of octaves and the group $G_2$, Nederl. Akad. Wetensch. Indag. Math. 21 (1959), 406-418. MR 27:2533
  • [G] J. van Geel, Applications of the Riemann-Roch theorem for curves to quadratic forms and division algebras, UCL, Recherches de math. 7 (1991), 1-32.
  • [H] R. Hartshorne, Algebraic Geometry, Springer-Verlag, New York-Heidelberg-Berlin, 1977. MR 57:3116
  • [J] N. Jacobson, Composition algebras and their automorphisms, Rend. Circ. Mat. Palermo 7 (1958), 55-80. MR 21:66
  • [Kn] M. Knebusch. Grothendieck- und Wittringe von nichtausgearteten symmetrischen Bilinearformen, Sitzungsber. Heidelb. Akad. Wiss. Math.-Natur. Kl., Springer-Verlag, Berlin-Heidelberg-New York, 1970. MR 42:6001
  • [KPS] M.-A. Knus, R. Parimala and R. Sridharan, On compositions and triality, J. Reine Angew. Math. 457 (1994), 45-70. MR 96a:11037
  • [Kü] B. Küting, Kompositionsalgebren über rationalen Funktionenkörpern, Dissertation. Fern Universität, Hagen, 1987.
  • [L] T. Y. Lam, Serre's Conjecture, Lecture Notes in Mathematics, vol. 635, Springer-Verlag, Berlin-Heidelberg-New York, 1978. MR 58:5644
  • [M] K. McCrimmon, Nonassociative algebras with scalar involution, Pacific J. Math. 116 (1995), 85-108. MR 86d:17003
  • [Mi] J. S. Milne, Étale Cohomology, Princeton University Press, Princeton, N.J., 1980. MR 81j:14002
  • [P1] H. P. Petersson, Composition algebras over algebraic curves of genus zero, Trans. Amer. Math. Soc. 337 (1993), 473-491. MR 93g:17006
  • [P2] H. P. Petersson, Composition algebras over a field with a discrete valuation, J. Algebra 29 (1974), 414-426. MR 51:635
  • [Pf] A. Pfister, Quadratic lattices in function fields of genus $0$, Proc. London Math. Soc. 66 (1993), 257-278. MR 94c:11030
  • [Pu1] S. Pumplün, Composition algebras over open dense subschemes of curves of genus zero, Non-Associative Algebra and its Applications (Santos Gonzalez, ed.). Kluwer Academic Publishers, Dordrecht, 1994, pp. 341-343. CMP 95:14
  • [Pu2] S. Pumplün, Kompositionsalgebren über Ringen vom Geschlecht Null. Dissertation, Fern Universität, Hagen, 1995.
  • [R] I. Reiner, Maximal Orders, Academic Press, London-New York-San Francisco, 1975. MR 52:13910
  • [S] W. Scharlau, Quadratic and Hermitian Forms, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1985. MR 86k:11022
  • [T] A. Tillmann, Unzerlegbare Vektorbündel über algbraischen Kurven, Dissertation. Fern Universität, Hagen, 1983.
  • [Wi] E. Witt, Über ein Gegenbeispiel zum Normensatz, Math. Z. 39 (1934), 462-467.

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Additional Information

S. Pumplün
Affiliation: Fakultät für Mathematik, Universität Regensburg, Universitätsstr. 31, 93040 Regensburg, Germany
Address at time of publication: Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico 88003
Email: susanne.pumpluen@mathematik.uni-regensburg.de

DOI: https://doi.org/10.1090/S0002-9947-99-01935-2
Received by editor(s): February 20, 1996
Article copyright: © Copyright 1999 American Mathematical Society

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