# Hochschild cohomology related to graded down-up algebras with weights $(1,n)$

## TL;DR

This paper computes Hochschild cohomology for graded down-up algebras with weights (1,n) for n ≥ 2, revealing how the derived category structure varies with algebra parameters and weights.

## Contribution

It extends Hochschild cohomology calculations to the case n ≥ 2 for graded down-up algebras, showing how derived categories differ based on algebraic conditions.

## Key findings

- Hochschild cohomology groups are explicitly calculated for n ≥ 2.
- Derived category structures depend on specific algebraic matrix conditions.
- Differences in Grothendieck groups are observed between cases n=2 and n≥3.

## Abstract

Let $A=A(\alpha, \beta)$ be a graded down-up algebra with $({\rm deg}\,x, {\rm deg}\,y)=(1,n)$ and $\beta \neq 0$, and let $\nabla A$ be the Beilinson algebra of $A$. If $n=1$, then a description of the Hochschild cohomology group of $\nabla A$ is known. In this paper, we calculate the Hochschild cohomology group of $\nabla A$ for the case $n \geq 2$. As an application, we see that the structure of the bounded derived category of the noncommutative projective scheme of $A$ is different depending on whether $\left(\begin{smallmatrix} 1&0 \end{smallmatrix}\right)\left(\begin{smallmatrix} \alpha &1 \\ \beta &0 \end{smallmatrix}\right)^n\left(\begin{smallmatrix} 1 \\ 0 \end{smallmatrix}\right)$ is zero or not. Moreover, it turns out that there is a difference between the cases $n=2$ and $n\geq 3$ in the context of Grothendieck groups.