Mathematical Journal of Okayama University volume63 issue1
Department of Mathematics, faculty of Science, Tokyo University of Science
Graduate School of Science and Technology, Shizuoka University
Classiﬁcation of AS-regular algebras is one of the main interests in non-commutative algebraic geometry. Recently, a complete list of superpotentials (deﬁning relations) of all 3-dimensional AS-regular algebras which are Calabi-Yau was given by Mori-Smith (the quadratic case) and Mori-Ueyama (the cubic case), however, no complete list of deﬁning relations of all 3-dimensional AS-regular algebras has not appeared in the literature. In this paper, we give all possible deﬁning relations of 3-dimensional quadratic AS-regular algebras. Moreover, we classify them up to isomorphism and up to graded Morita equivalence in terms of their deﬁning relations in the case that their point schemes are not elliptic curves. In the case that their point schemes are elliptic curves, we give conditions for isomorphism and graded Morita equivalence in terms of geometric data.
Mathematics Subject Classiﬁcation. 16W50, 16S37, 16D90, 16E65.