$\mathbb{Z}^2$-algebras as noncommutative blow-ups

TL;DR

This paper proves that certain well-behaved $Z^2$-algebras have categories equivalent to those of diagonal-like subalgebras, linking noncommutative birational transformations with noncommutative blow-ups.

Contribution

It establishes an equivalence between categories of $Z^2$-algebras and diagonal-like subalgebras, connecting noncommutative birational transformations with noncommutative blow-ups.

Findings

$Z^2$-algebras are QGr-equivalent to diagonal-like subalgebras.

These subalgebras serve as noncommutative blow-ups of Sklyanin algebras.

Links between noncommutative birational transformations and blow-ups are demonstrated.

Abstract

The goal of this note is to first prove that for a well behaved $\mathbb{Z}^2$-algebra $R$, the category $QGr(R) := Gr(R)/Tors(R)$ is equivalent to $QGr(R_\Delta)$ where $R_\Delta$ is a diagonal-like sub-$\mathbb{Z}$-algebra of $R$. Afterwards we use this result to prove that the $\mathbb{Z}^2$-algebras as introduced in [ArXiV:1607.08383] are QGr-equivalent to a diagonal-like sub-$\mathbb{Z}$-algebra which is a simultaneous noncommutative blow-up of a quadratic and a cubic Sklyanin algebra. As such we link the noncommutative birational transformation and the associated $\mathbb{Z}^2$-algebras as appearing in the work of Van den Bergh and Presotto with the noncommutative blowups appearing in the work of Rogalski, Sierra and Stafford.