Quantum projective planes finite over their centers

Abstract

For a 3-dimensional quantum polynomial algebra 𝐴 = A (𝐸 , 𝜎), Artin-Tate-Van den Bergh showed that 𝐴 is finite over its center if and only if | 𝜎 | < ∞. Moreover, Artin showed that if 𝐴 is finite over its center and 𝐸 ≠ 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 ≠ P 2 , then 𝐴 has a fat point module if and only if the quantum projective plane Proj nc 𝐴 is finite over its center in the sense of this paper if and only if |𝜈 * 𝜎 3 | < ∞ where 𝜈 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.

Twisted algebras of geometric algebras

“…) is a three-dimensional quadratic AS-regular algebra of Types T, T', and CC (so that |σ| = ∞ (cf. [9])…”

Twisted algebras of geometric algebras

“…Set τ := σ 2 ∈ Aut k E and S := A(E, σ 3 ). Since||τ || = ||σ 2 || = |σ 6 | = 1 by[9, Theorem 3.4], τ ∈ Aut k (E ↑ P 2 ). Since τ i+1 σ = σ 2i+3 = σ 3 τ i for every i ∈ Z, GrMod A ∼ = GrMod S by Theorem 2.6.…”