Lifting systems for finite length modules

Abstract

This paper is concerned with lifting modules along a surjective map of noetherian local rings, say $\ddot{I}†:R\twoheadrightarrow S$. A finitely generated $R$-module $L$ is a naive lift of an $S$-module $M$ if $L{\otimes }_{R}S\cong M$. We are concerned with the maximum depth and dimension among all naive lifts of $M$, which we call the liftable depth and liftable dimension, respectively, of $M$ along $\ddot{I}†$. We approach this via a notion of lifting systems that we introduce in this paper. We then provide a necessary and sufficient condition for a module of finite length to lift and Serre lift to a regular local ring in terms of lifting systems.