I will explain how to extend the theory of almost perfect domains to arbitrary Noetherian commutative rings R of Krull dimension 1, by showing that an R-module is cotorsion if and only if it is Ext^1-orthogonal to S^{-1}R, where S is the complement to the union of all minimal prime ideals in R.
The proof uses the theory of contramodules over a commutative ring with a fixed ideal, and the two-term complex R-->S^{-1}R in place of the traditional quotient module S^{-1}R/R.