Hochschild cohomology of Beilinson algebras of graded down-up algebras with weights ($n,m$) (2511.20195v1)

Abstract: Let $A=A(α, β)$ be a graded down-up algebra with weights $({\rm deg}\, x, {\rm deg}\, y)=(n,m)$ and $β\neq 0$, 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 $m\geq n\geq 1$.