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Comparison morphisms between two projective resolutions of monomial algebras
Volume 59, no. 1
(2018),
pp. 1–31
https://doi.org/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 $\operatorname{HH} ^
{\ast} (A)$.
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Published by the Unión Matemática Argentina |
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