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arXiv · 1905.02502

AS-regularity of geometric algebras of plane cubic curves

Ayako Itaba·Masaki Matsuno
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Abstract

Let $k$ be an algebraically closed field of characteristic $0$ and $A$ a graded $k$-algebra finitely generated in degree $1$. In this paper, for $3$-dimensional quadratic AS-regular algebras except for Type EC, we give a complete list of twisted superpotentials and a complete list of superpotentials such that derivation-quotient algebras are $3$-dimensional quadratic Calabi-Yau AS-regular algebras. For an algebra $A$ of Type EC, we give a criterion when $A$ is AS-regular. As an application, for an algebra $A$ of any type, we show that there exists a Calabi-Yau AS-regular algebra $S$ such that $A$ and $S$ are graded Morita equivalent. This result tells us that, for a $3$-dimensional quadratic AS-regular algebra $A$, to study the noncommutative projective scheme for $A$ defined by Artin-Zhang is reduced to study the noncommutative projective scheme for $S$ for the Calabi-Yau AS-regular algebra $S$.

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Ayako Itaba, Masaki Matsuno. 2019-05-07. AS-regularity of geometric algebras of plane cubic curves. https://doi.org/10.1017/s1446788721000070

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