The Cohomology Ring of Truncated Quiver Algebras

In this paper we determine the ring structure of the Hochschild cohomology of truncated quiver algebras with the Yoneda product. On the one hand Locateli described the cohomology groups in terms of classes of pairs of paths using minimal resolutions. On the other hand, the Yoneda product has a nice description on the bar resolution as the usual cup product. Our first main result is the explicit construction of comparison morphisms, in both directions, between the two different resolutions. We are then able to give a clear description of the Yoneda product on the minimal resolution and determine completely the structure of the cohomology ring. As a corollary we prove, that the product of two cohomology classes of odd degree is equal to zero. Moreover, we show that the ring structure (of the augmentation ideal) is trivial if the underlying quiver is not an oriented cycle and has neither sinks nor sources.