AS-REGULARITY of GEOMETRIC ALGEBRAS of PLANE CUBIC CURVES

Ayako Itaba, Masaki Matsuno

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1 Citation (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)193-217
Number of pages25
JournalJournal of the Australian Mathematical Society
Volume112
Issue number2
DOIs
Publication statusPublished - 22 Apr 2022

Keywords

  • AS-regular algebras
  • Calabi–Yau algebras
  • Koszul algebras
  • elliptic curves
  • geometric algebras
  • superpotentials

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