Abstract
This thesis is concerned with the problem of computing projective resolutions of associative algebras. Our starting point is Bardzell's resolution for monomial algebras. Given an associatve algebra, we use Bergman's principle of reduction systems to associate monomial algebras to it. We prove that the differentials in Bardzell's resolution of these monomial algebras can be modified to obtain projective resolutions of the original algebra. We also give sufficient conditions for a complex coming from a modification of Bardzell's resolution of an associated monomial algebra to be exact. We apply our method to three families of algebras: Quantum complete intersections, Quantum generalized Weyl algebras and down-up algebras. In the case of down-up algebras, we use the resolution obtained to compute homological invariants of these algebras. This way we prove regularity properties and we solve the isomorphism problem for non-noetherian down-up algebras.