On liftable and weakly liftable modules

Let T be a Noetherian ring and f a nonzerodivisor on T. We study concrete necessary and sufficient conditions for a module over R=T/(f) to be weakly liftable to T, in the sense of Auslander, Ding, and Solberg. We focus on cyclic modules and obtain various positive and negative results on the lifting and weak lifting problems. For a module over T we define the loci for certain properties: liftable, weakly liftable, having finite projective dimension and study their relationships.