Generalized Koszul Resolution

Abstract

In this paper, we generalize a notion of Koszul resolutions and characterize modules which admits such resolutions. It turns out that for a noetherian ring $A$ and a coherent $A$-module $M$, $M$ has a two dimensional generalized Koszul resolution if and only if $M$ is a pure weight two module in the sense of [HM10]. As an application, we attack the Gersten conjecture for weight two case.