Resolutions over Koszul algebras

Abstract

In this paper we show that if $Λ=\amalg_{i\geq 0}Λ_i$ is a Koszul algebra with $Λ_0$ isomorphic to a product of copies of a field, then the minimal projective resolution of $Λ_0$ as a right $Λ$-module provides all the information necessary to construct both a minimal projective resolution of $Λ_0$ as a left $Λ$-module and a minimal projective resolution of $Λ$ as a right module over the enveloping algebra of $Λ$. The main tool for this is showing that there is a comultiplicative structure on a minimal projective resolution of $Λ_0$ as a right $Λ$-module.