Lifting systems for finite length modules
Benjamin Katz, Nawaj KC, Kesavan Mohana Sundaram, Andrew J. Soto Levins, Ryan Watson
TL;DR
The paper introduces lifting systems as a practical framework to study lifting finite-length $S$-modules along a surjection $\varphi: R \twoheadrightarrow S$. It defines liftable depth and liftable dimension and proves two main results: a Tor-vanishing criterion that yields lower bounds on liftable depth, and a Fitting-ideal growth criterion that yields lower bounds on liftable dimension, with sharp connections to Serre liftability. It then demonstrates the utility of these concepts through explicit constructions, including unliftable finite-length modules that are Serre liftable and criteria ensuring Serre liftability for cyclic modules, thereby elucidating when lifts to regular rings exist or fail. The results advance understanding of how depth and dimension behave under lifts, relate to Hilbert–Samuel growth via Fitting ideals, and have implications for intersection properties of cyclic modules in regular settings.
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.