Comparison morphisms and the Hochschild cohomology ring of truncated quiver algebras

Abstract

A main contribution of this paper is the explicit construction of comparison morphisms between the standard bar resolution and Bardzell's minimal resolution for truncated quiver algebras (TQA's). As a direct application we describe explicitely the Yoneda product and derive several results on the structure of the cohomology ring of TQA's. For instance, we show that the product of odd degree cohomology classes is always zero. We prove that TQA's associated with quivers with no cycles or with neither sinks nor sources have trivial cohomology rings. On the other side we exhibit a fundamental example of a TQA with non trivial cohomology ring. Finaly, for truncated polyniomial algebras in one variable, we construct explicit cohomology classes in the bar resolution and give a full description of their cohomology ring.