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

Abstract

Let $A=A(\alpha ,\beta )$ be a graded down-up algebra with $(\mathrm{deg}x,\mathrm{deg}y)=(1,n)$ and $\beta \ne 0$, 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 $n\ge 2$. 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 $\left(\begin{array}{cc}1& 0\end{array}\right){\left(\begin{array}{cc}\alpha & 1\\ \beta & 0\end{array}\right)}^n\left(\begin{array}{c}1\\ 0\end{array}\right)$ is zero or not. Moreover, it turns out that there is a difference between the cases $n=2$ and $n\ge 3$ in the context of Grothendieck groups.

13 pages

References in corpus (1)

• Hochschild cohomology of noncommutative planes and quadrics

Cited by in corpus (1)

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