Comparison morphisms between two projective resolutions of monomial algebras

REDONDO, MARIA JULIA; ROMAN, LUCRECIA

REVISTA DE LA UNIóN MATEMáTICA ARGENTINA

UNION MATEMATICA ARGENTINA

Lugar: Bahia Blanca; Año: 2018 vol. 59 p. 1 - 31

0041-6932

We construct comparison morphisms between two well-known projective resolutions of a monomial algebra A: the bar resolution BarA and Bardzell´s resolution ApA.  The first one is used to define the cup product and the Lie bracket on the Hochschildcohomology HH∗(A) and the second one has been shown to be an efficient tool for computations 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, if A = kQ/I is a monomial algebra such that dimk eiAej = 1 whenever there exists an arrow α : i → j ∈ Q1, we describe the Lie action of the Lie algebra HH1(A) on HH∗(A).