Hochschild cohomology of Beilinson algebras of graded down-up algebras with weights ($n,m$)

TL;DR

This paper computes the Hochschild cohomology of Beilinson algebras associated with graded down-up algebras with weights, revealing new cases and their implications for derived categories of non-commutative schemes.

Contribution

It provides the Hochschild cohomology dimensional formula for cases where both weights are at least 2, extending previous results and analyzing derived category properties.

Findings

Hochschild cohomology dimensions are explicitly calculated for new weight cases.

Derived categories of certain non-commutative schemes are shown not to be equivalent to smooth projective surfaces.

The ring structure of Hochschild cohomology is described via the Yoneda product.

Abstract

Let $A=A(\alpha, \beta)$ be a graded down-up algebra with weights $({\rm deg}\, x, {\rm deg}\, y)=(n,m)$ and $\beta \neq 0$, and $\nabla A$ the Beilinson algebra of $A$. Note that $A$ is a $3$-dimensional cubic AS-regular algebra. Assume that $\gcd(n, m)=1$ and $m \geq n$. If $n=1$ and $m=1$, then a description of the Hochschild cohomology group of $\nabla A$ was already known by Belmans. If $n=1$ and $m \geq 2$, then the dimensional formula of the Hochschild cohomology group of $\nabla A$ was given by the first author and Ueyama. In this paper, we give the dimensional formula of the Hochschild cohomology group of $\nabla A$ for the case that $n \geq 2$ and $m \geq 2$. As a byproduct of this dimensional formula, we prove that, for $m>n>1$, the derived category of a non-commutative projective scheme associated to $A$ is not equivalent to the derived category of any smooth projective surface. Moreover, we give the ring structure on the Hochschild cohomology group with the Yoneda product for the case that $m\geq n\geq 1$.