Projective resolutions of associative algebras and ambiguities

The aim of this article is to give a method to construct bimodule resolutions of associative algebras, generalizing Bardzell’s well-known resolution of monomial algebras. We stress that this method leads to concrete computations, providing thus a useful tool for computing invariants associated to the algebras. We illustrate how to use it giving several examples in the last section of the article. In particular we give necessary and sufficient conditions for noetherian down-up algebras to be 3-Calabi-Yau.