Minimal Resolutions of Algebras

A method is described for constructing the minimal projective resolution of an algebra considered as a bimodule over itself. The method applies to an algebra presented as the quotient of a tensor algebra over a separable algebra by an ideal of relations that is either homogeneous or admissible (with some additional finiteness restrictions in the latter case). In particular, it applies to any finite-dimensional algebra over an algebraically closed field. The method is illustrated by a number of examples, viz. truncated algebras, monomial algebras, and Koszul algebras, with the aim of unifying existing treatments of these in the literature.