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

Abstract

Let $A=A(\alpha, \beta)$ be a graded down-up algebra with weights $(\mathrm{deg}\, x, \mathrm{deg}\, y)=(n,m)$ and $\beta \neq 0$, and $\nabla A$ its Beilinson algebra. Such an algebra $A$ is a 3-dimensional cubic AS-regular algebra by Kirkman--Musson--Passman. Assuming $\mathrm{gcd}\,(n, m)=1$ and $m \geq n$, we extend the previous results on the Hochschild cohomology of $\nabla A$. Known cases include $(n,m) = (1,1)$ (Belmans) and $(n = 1,\,m \geq 2)$ (Itaba--Ueyama). In this paper, we determine the dimensions of the Hochschild cohomology groups of $\nabla A$ in the remaining case $n\geq 2$ and $m\geq 2$ by explicitly constructing the projective resolution and computing the ranks of the arising representation matrices. As a byproduct, for $m>n>1$, we show that the derived category of the noncommutative projective scheme associated to $A$ is not equivalent to the derived category of any smooth projective surface. Moreover, for all $m \geq n \geq 1$, we describe the ring structure of the Hochschild cohomology group $\nabla A$ with respect to the Yoneda product.