Hochschild cohomology of Beilinson algebras of graded down-up algebras with weights ($n,m$)
Ayako Itaba, Shu Minaki
TL;DR
The paper analyzes the Hochschild cohomology of Beilinson algebras $\nabla A$ attached to graded down-up AS-regular algebras $A(\alpha,\beta)$ with weights $(n,m)$ under $\gcd(n,m)=1$, $m\ge n$, and $\beta\neq0$. It computes the dimensions of $HH^0$, $HH^1$, and $HH^2$ of $\nabla A$ and shows $HH^r=0$ for $r\ge3$, with dimensions depending on the parity of $n+m$ and the parameter $\alpha$; the method follows a Green–Snashall-type resolution and representation-matrix analysis. A key consequence is that for $m>n>1$, the derived category of tails $A$ is not equivalent to the derived category of any smooth projective surface, highlighting a noncommutative-geometry obstruction. The authors also determine the ring structure of $\bigoplus_{r\ge0} HH^r(\nabla A)$ by presenting it as a quotient of an exterior algebra $\Lambda(a,b)$ by an explicit homogeneous ideal, with detailed generators provided in various cases. Together, these results offer precise invariants and a full ring-theoretic picture of Hochschild cohomology for this broad class of Beilinson algebras, contributing to noncommutative projective geometry and derived-category comparisons.
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 $m\geq n\geq 1$.