Comparison morphisms between two projective resolutions of monomial algebras

María Julia Redondo,Lucrecia Román

doi:10.33044/revuma.v59n1a01

Abstract

We construct comparison morphisms between two well-known projective resolutions of a monomial algebra $A$: the bar resolution $\operatorname{\mathbb{Bar}} A$ and Bardzell's resolution $\operatorname{\mathbb{Ap}} A$; the first one is used to define the cup product and the Lie bracket on the Hochschild cohomology $\operatorname{HH} ^*(A)$ and the second one has been shown to be an efficient tool for computation of these cohomology groups. The constructed comparison morphisms allow us to show that the cup product restricted to even degrees of the Hochschild cohomology has a very simple description. Moreover, for $A= \mathbb{k} Q/I$ a monomial algebra such that $\dim_ \mathbb{k} e_i A e_j = 1$ whenever there exists an arrow $\alpha: i \to j \in Q_1$, we describe the Lie action of the Lie algebra $\operatorname{HH}^1(A)$ on $\operatorname{HH}^{\ast} (A)$.

Highlights

Hochschild cohomology of associative algebras was introduced by G
In the case where A is an algebra over a ring k such that A is k-projective, Cartan and Eilenberg give a useful interpretation of the Hochschild cohomology groups HHn(A, A) with coefficients in A
Despite this freedom of choice to calculate cohomology groups, it is not the case with some of the structures defined in cohomology: the sum HH∗(A) = ⊕n≥0HHn(A) is a Gerstenhaber algebra, that is, it is a graded commutative ring via the cup product ∪ : HHn(A) × HHm(A) → HHn+m(A), a graded Lie algebra via the bracket [−, −] : HHn(A) × HHm(A) → HHn+m−1(A), and these two structures are related, see [7]

Summary

Introduction

Hochschild cohomology of associative algebras was introduced by G. In the case where A is an algebra over a ring k such that A is k-projective, Cartan and Eilenberg give a useful interpretation of the Hochschild cohomology groups HHn(A, A) with coefficients in A They prove that these groups can be identified with the groups ExtnAe (A, A) and be calculated using arbitrary projective resolutions of A over its enveloping algebra Ae, see [5]. Important improvements have been made when considering the cup product: it has another description, the Yoneda product, which can be transported to the complex obtained by using any other projective resolution This has been used to describe the ring structure of HH∗(A) for many algebras A such as radical square zero algebras [6], truncated quiver algebras [1], Koszul algebras [4] and so on. We show that the cup product restricted to even degrees of the Hochschild cohomology has a very simple description

Preliminaries

The comparison morphisms

The map F is a comparison morphism

The map G is a comparison morphism