On liftings of modules of finite projective dimension

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.

Significant changes in this version in line with the suggestions from the referee. The name "Dimly liftable" from the previous version has now been changed to "Serre liftable". To appear in Int. Math. Res. Not. IMRN. 8 pages