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 over arbitrary fields (TQA's). As a direct application we describe explicitly the Yoneda product and derive several results on the structure of the cohomology ring of TQA's over a field of characteristic zero. 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 nontrivial cohomology ring. Finally, for truncated polynomial algebras in one variable, we construct explicit cohomology classes in the bar resolution and give a full description of their cohomology ring.

Hochschild cohomology rings of d-Koszul algebras

“…, f d are cohomology classes of positive degree, then the product f 1 · · · f d = 0. In particular, HH * (A)/N ∼ = K (where N is the ideal generated by homogeneous nilpotent elements) and hence the Snashall-Solberg conjecture holds true for truncated quiver algebras [1]. Furthermore, G. Ames, L. Cagliero and P. Tirao completely determined the multiplicative structure of Hochschild cohomology rings of two large classes of truncated quiver algebras, see [1] for details.…”

“…Hochschild homology and cohomology of truncated quiver algebras have been extensively studied by many authors [1,2,5,12,16,15,20,22].…”

“…Solberg described the multiplicative structure of Hochschild cohomology rings of Koszul algebras based on a ''comultiplicative'' structure of a minimal projective bimodule resolution [10]. For the case of monomial d-Koszul algebras, G. Ames, L. Cagliero and P. Tirao gave a clear description of the Yoneda product and completely determined the structure of Hochschild cohomology rings of truncated quiver algebras by the explicit constructions of comparison morphisms between the two different resolutions [1]. In [19], Siegel and Witherspoon defined a cup product of two elements η in HH n (A) and θ in HH m (A) using the composition…”

“…Finally, we apply our results to monomial d-Koszul algebras and the well-known cubic Artin-Schelter regular algebra of global dimension 3 of type A in Section 4. As a consequence, we reobtain the description of the multiplicative structure of Hochschild cohomology rings of truncated quiver algebras in a different way [1].…”

Rigid quivers and rigid algebras

“…In contrast, if Δ is not a crown all classes of parallel paths of different lengths have an extreme. It follows in particular that if Δ has neither sinks nor sources, and it is not a crown, then it has no medals at all (see [ACT,§8]). The following two examples illustrate this definition and turn out to be very relevant in the classification of medals carried out in Section 5.…”