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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Algebras associated to elliptic curves


Author: Darin R. Stephenson
Journal: Trans. Amer. Math. Soc. 349 (1997), 2317-2340
MSC (1991): Primary 16W50, 14A22, 16P40; Secondary 16P90, 16E10
MathSciNet review: 1390046
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Abstract: This paper completes the classification of Artin-Schelter regular algebras of global dimension three. For algebras generated by elements of degree one this has been achieved by Artin, Schelter, Tate and Van den Bergh. We are therefore concerned with algebras which are not generated in degree one. We show that there exist some exceptional algebras, each of which has geometric data consisting of an elliptic curve together with an automorphism, just as in the case where the algebras are assumed to be generated in degree one.

In particular, we study the elliptic algebras $A(+)$, $A(-)$, and $A({\mathbf {a}})$, where ${\mathbf {a}}\in \mathbb {P}^{2}$, which were first defined in an earlier paper. We omit a set $S\subset \mathbb {P}^2$ consisting of 11 specified points where the algebras $A({\mathbf {a}})$ become too degenerate to be regular.

Theorem. Let $A$ represent $A(+)$, $A(-)$ or $A({\mathbf {a}})$, where ${\mathbf {a}} \in \mathbb {P}^2\setminus S$. Then $A$ is an Artin-Schelter regular algebra of global dimension three. Moreover, $A$ is a Noetherian domain with the same Hilbert series as the (appropriately graded) commutative polynomial ring in three variables.

This, combined with our earlier results, completes the classification.


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Additional Information

Darin R. Stephenson
Affiliation: Department of Mathematics, University of California, San Diego, San Diego, California 92093
Email: dstephen@math.ucsd.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-97-01769-8
PII: S 0002-9947(97)01769-8
Keywords: Graded ring, Artin-Schelter regular algebra, noncommutative projective geometry, elliptic algebra, point module
Received by editor(s): November 14, 1995
Additional Notes: This research was supported in part by a graduate research fellowship on NSF grant number 9304423.
Article copyright: © Copyright 1997 American Mathematical Society