Comparison morphisms and the Hochschild cohomology ring of truncated quiver algebras

G. AMES; L. CAGLIERO; P. TIRAO

JOURNAL OF ALGEBRA

Elsevier

Año: 2009 p. 1466 - 1497

0021-8693

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.