Lifting systems for finite length modules

TL;DR

This paper investigates the conditions under which modules over noetherian local rings can be lifted along surjective ring maps, introducing lifting systems to analyze liftable depth and dimension.

Contribution

It introduces the concept of lifting systems and provides necessary and sufficient conditions for finite length modules to lift and Serre lift to regular local rings.

Findings

Characterization of liftable modules via lifting systems

Necessary and sufficient conditions for finite length modules to lift

Analysis of liftable depth and dimension along ring surjections

Abstract

This paper is concerned with lifting modules along a surjective map of noetherian local rings, say $\varphi \colon R \twoheadrightarrow S$. 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 $\varphi$. 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.