Quantum Projective Planes Finite over their Centers

Abstract

For a $3$-dimensional quantum polynomial algebra $A=\mathcal{A}(E,\sigma )$, Artin-Tate-Van den Bergh showed that $A$ is finite over its center if and only if $|\sigma |<\infty$. Moreover, Artin showed that if $A$ is finite over its center and $E\ne {\mathbb{P}}^2$, then $A$ has a fat point module, which plays an important role in noncommutative algebraic geometry, however the converse is not true in general. In this paper, we will show that, if $E\ne {\mathbb{P}}^2$, then $A$ has a fat point module if and only if the quantum projective plane ${\mathsf{Proj}}_{\mathrm{nc}}A$ is finite over its center in the sense of this paper if and only if $|{\nu }^{\ast }{\sigma }^3|<\infty$ where $\nu$ is the Nakayama automorphism of $A$.In particular, we will show that if the second Hessian of $E$ is zero, then $A$ has no fat point module.

13 pages

References in corpus (3)

• Compact moduli of noncommutative projective planes
• AS-regularity of geometric algebras of plane cubic curves
• A complete classification of $3$-dimensional quadratic AS-regular algebras of Type EC

Cited by in corpus (1)

• Noncommutative conics in Calabi-Yau quantum projective planes