On liftings of modules of finite projective dimension

TL;DR

This paper introduces Serre liftable modules and demonstrates their significance in proving cases of Serre's positivity conjecture and the Length Conjecture, linking module liftability to ring invariants.

Contribution

It defines Serre liftable modules and proves new cases of Serre's positivity conjecture and the Length Conjecture using these modules.

Findings

Proved Serre's positivity conjecture for Serre liftable modules over ramified regular local rings.

Established a lower bound on the length of Serre liftable modules using Hilbert-Samuel multiplicity.

Linked module liftability to key conjectures in commutative algebra.

Abstract

We introduce and study a notion of Serre liftable modules; these are modules that are liftable to modules of the maximal possible dimension over a regular local ring. We establish new cases of Serre's positivity conjecture over ramified regular local rings by proving it for Serre liftable modules. Furthermore, we show that the length of a nonzero Serre liftable module is bounded below by the Hilbert-Samuel multiplicity of the local ring. This establishes special cases of the Length Conjecture of Iyengar-Ma-Walker.