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

Let $A=A(\alpha ,\beta )$ be a graded down-up algebra with weights $(\mathrm{d}\mathrm{e}\mathrm{g}x,\mathrm{d}\mathrm{e}\mathrm{g}y)=(n,m)$ and $\beta \mathrm{\ne }0$, and $\mathrm{\nabla }A$ its Beilinson algebra. Such an algebra $A$ is a 3-dimensional cubic AS-regular algebra by Kirkman–Musson–Passman. Assuming $\mathrm{g}\mathrm{c}\mathrm{d}(n,m)=1$ and $m\ge n$, we extend the previous results on the Hochschild cohomology of $\mathrm{\nabla }A$. Known cases include $(n,m)=(1,1)$ (Belmans) and $(n=1,m\ge 2)$ (Itaba–Ueyama). In this paper, we determine the dimensions of the Hochschild cohomology groups of $\mathrm{\nabla }A$ in the remaining case $n\ge 2$ and $m\ge 2$ 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 $m\ge n\ge 1$, we describe the ring structure of the Hochschild cohomology group $\mathrm{\nabla }A$ with respect to the Yoneda product.