Projective resolutions of associative algebras and ambiguities

TL;DR

This paper introduces a new method for constructing bimodule resolutions of associative algebras, generalizing existing techniques, and demonstrates its effectiveness through examples and applications to noetherian down-up algebras.

Contribution

It generalizes Bardzell's resolution for monomial algebras to a broader class, enabling concrete computations of algebra invariants.

Findings

Provides a constructive method for bimodule resolutions

Offers necessary and sufficient conditions for 3-Calabi-Yau property

Includes multiple illustrative examples

Abstract

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.