Abstract
AbstractIn 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.
Highlights
In noncommutative algebraic geometry, an Artin–Schelter regular (AS-regular) algebra, introduced by Artin and Schelter [1], is one of the main interests
We give a complete list of superpotentials whose derivation-quotient algebras are three-dimensional quadratic Calabi–Yau AS-regular algebras whose point schemes are not elliptic curves
By using a twist of a superpotential, we show that the potentials listed in Proposition 3.1 are twisted superpotentials and their derivation-quotient algebras are three-dimensional quadratic AS-regular algebras
Summary
An Artin–Schelter regular (AS-regular) algebra, introduced by Artin and Schelter [1], is one of the main interests. Mori and Smith [14] classified three-dimensional quadratic Calabi–Yau AS-regular algebras by using superpotentials. For geometric algebras listed in [10, Theorem 3.1], we give a list of candidates of twisted superpotentials to serve our purposes (see Proposition 3.1) By using this list, we give a complete list of superpotentials whose derivation-quotient algebras are three-dimensional quadratic Calabi–Yau AS-regular algebras whose point schemes are not elliptic curves (see Theorem 3.3). Theorem 1.1 tells us that, for a three-dimensional quadratic AS-regular algebra A, the study of the noncommutative projective scheme ProjncA of A in the sense of Artin and Zhang [2] is reduced to the study of ProjncS for the Calabi–Yau AS-regular algebra S.
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