$S_r$ properties of generalized Cohen--Macaulay modules and rings having liftable local cohomology via Serre depth

We study Serre depth via Matlis duals of local cohomology modules. We relate the Serre depth of a module to that of a quotient by a regular element, characterize the $S_r$ property for generalized Cohen–Macaulay modules in terms of ordinary depth, and show that the $S_r$ property descends to reduced structures under liftable local cohomology hypotheses.