On liftings of modules of finite projective dimension

We introduce and study a notion of dimly liftable modules; these are modules that are liftable to the right dimension to a regular local ring. We establish special new cases of Serre's positivity conjecture over ramified regular local rings by proving it for dimly liftable modules. Furthermore, we show that the length of a nonzero finite length dimly 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.