A periodic projective bimodule resolution of an algebra associated with a cyclic quiver and a separable algebra, and the Hochschild cohomology ring

Abstract

Let $\mathrm{\Delta }$ be a separable algebra over a commutative ring $R$ and $f(x)$ a monic polynomial over the center of $\mathrm{\Delta }$. We deal with the $R$-algebra $\text{Λ}=\mathrm{\Delta }\text{Γ}/(f(X^s))$, where $\mathrm{\Delta }\text{Γ}$ is the path algebra of the cyclic quiver $\text{Γ}$ with $s$ vertices and $s$ arrows, and $X$ is the sum of all arrows. We show that $\text{Λ}$ has a periodic projective bimodule resolution of period 2. Moreover, by using the resolution, we describe the structure of the Hochschild cohomology ring of $\text{Λ}$ by means of the Yoneda product.