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AS-REGULARITY OF GEOMETRIC ALGEBRAS OF PLANE CUBIC CURVES

Part of: Rings and algebras with additional structure Rings and algebras arising under various constructions Homological methods Modules, bimodules and ideals

Published online by Cambridge University Press:  22 June 2021

AYAKO ITABA Open the ORCID record for AYAKO ITABA [Opens in a new window]  and
MASAKI MATSUNO Open the ORCID record for MASAKI MATSUNO [Opens in a new window]
AYAKO ITABA*
Affiliation:
Department of Mathematics, Faculty of Science, Tokyo University of Science, 1-3 Kagurazaka, Shinjyuku-ku, Tokyo162-8601, Japan and Institute of Arts and Sciences, Katsushika Division, Tokyo University of Science, 6-3-1 Niijjuku, Katsushika-ku, Tokyo125-8585, Japan
MASAKI MATSUNO
Affiliation:
Graduate School of Science and Technology, Shizuoka University, Ohya 836, Shizuoka422-8529, Japan e-mail: [email protected]
*
e-mail: [email protected]
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Abstract

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In noncommutative algebraic geometry an Artin–Schelter regular (AS-regular) algebra is one of the main interests, and every three-dimensional quadratic AS-regular algebra is a geometric algebra, introduced by Mori, whose point scheme is either $\mathbb {P}^{2}$ or a cubic curve in $\mathbb {P}^{2}$ by Artin et al. [‘Some algebras associated to automorphisms of elliptic curves’, in: The Grothendieck Festschrift, Vol. 1, Progress in Mathematics, 86 (Birkhäuser, Basel, 1990), 33–85]. In the preceding paper by the authors Itaba and Matsuno [‘Defining relations of 3-dimensional quadratic AS-regular algebras’, Math. J. Okayama Univ. 63 (2021), 61–86], we determined all possible defining relations for these geometric algebras. However, we did not check their AS-regularity. In this paper, by using twisted superpotentials and twists of superpotentials in the Mori–Smith sense, we check the AS-regularity of geometric algebras whose point schemes are not elliptic curves. For geometric algebras whose point schemes are elliptic curves, we give a simple condition for three-dimensional quadratic AS-regular algebras. As an application, we show that every three-dimensional quadratic AS-regular algebra is graded Morita equivalent to a Calabi–Yau AS-regular algebra.

Keywords

AS-regular algebrasCalabi–Yau algebraselliptic curvesgeometric algebrasKoszul algebrassuperpotentials

MSC classification

Primary: 16W50: Graded rings and modules
Secondary: 16S37: Quadratic and Koszul algebras 16D90: Module categories ; module theory in a category-theoretic context; Morita equivalence and duality 16E65: Homological conditions on rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 112 , Issue 2 , April 2022 , pp. 193 - 217
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© 2021 Australian Mathematical Publishing Association Inc.

Footnotes

Communicated by Daniel Chan

The first author was supported by Grants-in-Aid for Young Scientific Research 18K13397 Japan Society for the Promotion of Science.

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