TY - JOUR

T1 - AS-REGULARITY of GEOMETRIC ALGEBRAS of PLANE CUBIC CURVES

AU - Itaba, Ayako

AU - Matsuno, Masaki

N1 - Funding Information:
The authors thank the referee for helpful comments in improving the paper. They are grateful to Professor Izuru Mori for his support and helpful discussions. Also, they would like to thank Professor Kenta Ueyama for his helpful comments. Moreover, they appreciate Shinichi Hasegawa and Kosuke Shima for helping to build and to check Table 1 in Proposition 3.1. For Remark 3.2 (2), the authors thank Professor Andrew Conner and Professor Peter Goetz for informing them of a typo for the relation of Type TL4.
Publisher Copyright:
© 2022 Cambridge University Press. All rights reserved.

PY - 2022/4/22

Y1 - 2022/4/22

N2 - 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 or a cubic curve in 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.

AB - 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 or a cubic curve in 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.

KW - AS-regular algebras

KW - Calabi–Yau algebras

KW - Koszul algebras

KW - elliptic curves

KW - geometric algebras

KW - superpotentials

UR - http://www.scopus.com/inward/record.url?scp=85108831733&partnerID=8YFLogxK

U2 - 10.1017/S1446788721000070

DO - 10.1017/S1446788721000070

M3 - Article

AN - SCOPUS:85108831733

VL - 112

SP - 193

EP - 217

JO - Journal of the Australian Mathematical Society

JF - Journal of the Australian Mathematical Society

SN - 1446-7887

IS - 2

ER -