Approximations with modules having linear resolutions

Abstract

Let Λ be a Koszul algebra over a field K. We study in this paper a class of modules closely related to the Koszul modules called weakly Koszul modules. It turns out that these modules have some special filtrations with modules having linear resolutions and therefore easy to describe minimal projective resolutions. We prove that if the Koszul dual of a finite-dimensional Koszul algebra is Noetherian then every finitely generated graded module has a weakly Koszul syzygy and as a consequence a rational Poincaré series. If Λ is selfinjective Koszul, we prove that the stable part of each connected component of the graded Auslander–Reiten quiver containing a weakly Koszul module is of the form $\text{Z}\text{A}{}_{\mathrm{\infty }}$, and if the Koszul dual of Λ is Noetherian, then every component has its stable part of the form $\text{Z}\text{A}{}_{\mathrm{\infty }}$.