Comparison morphisms and the Hochschild cohomology ring of truncated quiver algebras

G. AMES, ; L. CAGLIERO,; P. TIRAO

JOURNAL OF ALGEBRA

Elsevier

Año: 2009 vol. 322 p. 1466 - 1497

0021-8693

In this paper we investigate the ring structure of the Hochschild cohomology ring of truncated quiver algebras.While there exists a description of the cohomology groups of these algebras in terms of minimal resolutions, the Yoneda product is well understood only at the level of the bar resolution.A main result of this paper is the explicit construction, for any truncated quiver algebra, of comparison morphisms between these two different resolutions, in both directions and for all degrees.We are then able to understand the Yoneda product both, for the complex associated to the minimal resolution, and also at the cohomology level.We prove that the cohomology ring is bigraded with respect to a natural bigrading of the complex constructed from the minimal resolution.We exhibit examples of non cycle truncated quiver algebras with non trivial Yoneda product. In contrast we prove that the Yoneda product is zero in positive cohomological degrees for two large classes of truncated quiver algebras.As a final application of the comparison morphism we produce,for general truncated quiver algebras, many explicit non zerocohomology classes in the bar resolution. In the particular case of the algebra of truncated polynomials in one variable, we exhibit a basis of the cohomology consisting of classes in the bar resolution.