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

Abstract

Let Δ be a separable algebra over a commutative ring R and f(x) a monic polynomial over the center of Δ. We deal with the R-algebra Λ=ΔΓ/(f(Xs)), where ΔΓ is the path algebra of the cyclic quiver Γ with s vertices and s arrows, and X is the sum of all arrows. We show that 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 Λ by means of the Yoneda product.