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Some Algebras Associated to Automorphisms of Elliptic Curves

  • Michael Artin
  • John Tate
  • M. Van den Bergh
Part of the Progress in Mathematics book series (MBC)

Abstract

The main object of this paper is to relate a certain type of graded algebra, namely the regular algebras of dimension 3, to automorphisms of elliptic curves. Some of the results were announced in [V]. A graded algebra A is called regular if it has finite global dimension, polynomial growth, and is Gorenstein. The precise definitions are reviewed in Section 2. As was shown in [A-S], there are two basic possibilities for a regular algebra A of (global) dimension 3 which is generated in degree 1. Either A can be presented by 3 generators and 3 quadratic relations, or else by 2 generators and 2 cubic relations. Throughout this paper, A will denote an algebra so presented, over a ground field k.

Keywords

Exact Sequence Irreducible Component Elliptic Curf Homogeneous Element Hilbert Function 
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References

  1. [An]
    D. Anick, On the homology of associative algebras, Trans. Amer. Math. Soc. 296 (1986), 641–659.MathSciNetCrossRefMATHGoogle Scholar
  2. [A-K]
    A. Altman and S. Kleiman, Introduction to Grothendteck Duality Theory, Lecture Notes in Math. 146 (1970).Google Scholar
  3. [A-T]
    E. Artin and J. Tate, A note on finite ring extensions, J. Math. Soc. Japan 3 (1951), 74–77.MathSciNetCrossRefGoogle Scholar
  4. [A-S]
    M. Artin and W. Schelter, Graded algebras of global dimension 3, Advances in Math 66 (1987), 171–216.MathSciNetGoogle Scholar
  5. [A-T-V]
    M. Artin, M., J. Tate, and M. Van Den Bergh, Modules over regular algebras of dimension 3, (in preparation).Google Scholar
  6. [Au]
    M. Auslander, On the dimension of modules and algebras III, Nagoya Math. J. 9 (1955), 67–77.MathSciNetCrossRefMATHGoogle Scholar
  7. [B1]
    N. Bourbaki, Algèbre, Eléments de Mathématiques, Hermann, Paris 1960–65.MATHGoogle Scholar
  8. [B2]
    N. Bourbaki, Algèbre Commutative, Eléments de Mathématiques, Hermann, Paris 1960–65.MATHGoogle Scholar
  9. [C]
    H. Cartan, Homologie et cohomologie d’une algèbre gradueé, Séminaire Cartan, 11e année, 57–58, exposé 15.Google Scholar
  10. [C-E]
    H. Cartan and S. Eilenberg, Homological Algebra, Princeton University Press, Princeton 1956.MATHGoogle Scholar
  11. [EGA]
    A. Grothendieck and J. Dieudonné, Eléments de géométrie algébrique, Pub. Math Inst. Hautes Études Sci. 1960–67.MATHGoogle Scholar
  12. [H]
    R. Hartshorne, Algebraic Geometry, Springer-Verlag, New York 1977.CrossRefMATHGoogle Scholar
  13. [O-F]
    Одесский, А. Б. and Б. Л. Фейгнн, Алгебры Склянин, ассоциированные с эллиптической кривой, (manuscript).Google Scholar
  14. [N-O]
    С. Nastacescu and R. Van Oystaeyen, Graded Ring Theory, North-Holland, Amsterdam 1982.Google Scholar
  15. [R]
    C. P. Ramanujam, Remarks on the Kodaira vanishing theorem, J. Indian Math. Soc. 36 (1972), 41–51.MathSciNetMATHGoogle Scholar
  16. [S]
    Склянин, Е. К., О некоторых алгебраических структурах, связанных с уравнением янга-Бакстера., II Представления квантовой алгебры, — Функцион анализ 17 (1983), 34–38.Google Scholar
  17. [W]
    Wall, C. T. C, Generators and relation for the Steenrod algebras, Annals of Math 72 (1960), 429–444.CrossRefMATHGoogle Scholar
  18. [V]
    Van Den Bergh, M., Regular algebras of dimenston 3, Séminaire Dubreil-Malliavin 1986, Lecture Notes in Math. 1296, Springer-Verlag, Berlin 1987, 228–234.Google Scholar

Copyright information

© Springer Science+Business Media New York 2007

Authors and Affiliations

  • Michael Artin
    • 1
  • John Tate
    • 2
  • M. Van den Bergh
    • 3
  1. 1.Dept. of MathematicsMITCambridgeUSA
  2. 2.Dept. of MathematicsHarvard UniversityCambridgeUSA
  3. 3.Dept. Wisk. en Inform.UIAWilrijkBelgium

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