Hochschild cohomology of truncated quiver algebras*

Abstract

Consider a truncated quiver algebra , where Γ is a finite quiverJ is the bilateral ideal of kΓ generated by the arrows, and k, is a field with characteristic zero. We give another description of a minimal projective resolution of Λ as a Λe-module given in [2] and [8], and using this description we compute the dimensions, as a vector space, of each Hochschild cohomology group of the truncated quiver algebra, in terms of its quiver Γ. We also prove that the Hochschild cohomology ring of Λ is finite dimensional as a k-vector space if and only if Γ has no oriented cycles.