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.
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References
D. Anick, On the homology of associative algebras, Trans. Amer. Math. Soc. 296 (1986), 641–659.
A. Altman and S. Kleiman, Introduction to Grothendteck Duality Theory, Lecture Notes in Math. 146 (1970).
E. Artin and J. Tate, A note on finite ring extensions, J. Math. Soc. Japan 3 (1951), 74–77.
M. Artin and W. Schelter, Graded algebras of global dimension 3, Advances in Math 66 (1987), 171–216.
M. Artin, M., J. Tate, and M. Van Den Bergh, Modules over regular algebras of dimension 3, (in preparation).
M. Auslander, On the dimension of modules and algebras III, Nagoya Math. J. 9 (1955), 67–77.
N. Bourbaki, Algèbre, Eléments de Mathématiques, Hermann, Paris 1960–65.
N. Bourbaki, Algèbre Commutative, Eléments de Mathématiques, Hermann, Paris 1960–65.
H. Cartan, Homologie et cohomologie d’une algèbre gradueé, Séminaire Cartan, 11e année, 57–58, exposé 15.
H. Cartan and S. Eilenberg, Homological Algebra, Princeton University Press, Princeton 1956.
A. Grothendieck and J. Dieudonné, Eléments de géométrie algébrique, Pub. Math Inst. Hautes Études Sci. 1960–67.
R. Hartshorne, Algebraic Geometry, Springer-Verlag, New York 1977.
Одесский, А. Б. and Б. Л. Фейгнн, Алгебры Склянин, ассоциированные с эллиптической кривой, (manuscript).
С. Nastacescu and R. Van Oystaeyen, Graded Ring Theory, North-Holland, Amsterdam 1982.
C. P. Ramanujam, Remarks on the Kodaira vanishing theorem, J. Indian Math. Soc. 36 (1972), 41–51.
Склянин, Е. К., О некоторых алгебраических структурах, связанных с уравнением янга-Бакстера., II Представления квантовой алгебры, — Функцион анализ 17 (1983), 34–38.
Wall, C. T. C, Generators and relation for the Steenrod algebras, Annals of Math 72 (1960), 429–444.
Van Den Bergh, M., Regular algebras of dimenston 3, Séminaire Dubreil-Malliavin 1986, Lecture Notes in Math. 1296, Springer-Verlag, Berlin 1987, 228–234.
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Artin, M., Tate, J., Van den Bergh, M. (2007). Some Algebras Associated to Automorphisms of Elliptic Curves. In: Cartier, P., Illusie, L., Katz, N.M., Laumon, G., Manin, Y.I., Ribet, K.A. (eds) The Grothendieck Festschrift. Progress in Mathematics. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-4574-8_3
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