On the regularity and Koszulness of modules over local rings

TL;DR

This paper investigates the properties of Koszul modules over Noetherian local rings, providing new characterizations and extending existing results using special filtrations, with implications for their homological behavior.

Contribution

It introduces a novel approach using special filtrations to characterize and extend classes of Koszul modules over local rings.

Findings

Large classes of modules over Koszul rings are Koszul.

Reproves and extends results by Fitzgerald.

Provides new characterizations of Koszul modules.

Abstract

Koszul modules over Noetherian local rings $R$ were introduced by Herzog and Iyengar and they possess good homological properties, for instance their Poincare' series is rational. It is an interesting problem to characterize classes of Koszul modules. Following the idea traced by Avramov, Iyengar and Sega, we take advantage of the existence of special filtration on $R$ for proving that large classes of $R$-modules over Koszul rings are Koszul modules. By using this tool we reprove and extend some results obtained by Fitzgerald.