Mathematical Journal of Okayama University volume63 issue1
2021-01 発行

Defining relations of 3-dimensional quadratic AS-regular algebras

Itaba, Ayako Department of Mathematics, faculty of Science, Tokyo University of Science
Matsuno, Masaki Graduate School of Science and Technology, Shizuoka University
Publication Date
2021-01
Abstract
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.
Keywords
AS-regular algebras
geometric algebras
quadratic algebras
nodal cubic curves
elliptic curves
Hesse form
Sklyanin algebras
Comments
Mathematics Subject Classification. 16W50, 16S37, 16D90, 16E65.