  In this paper, we consider the openness of the P-locus of a finitely
generated module over a commutative noetherian ring in the case where P is each
of the properties FID, Gor, CM, MCM, (S_n), and (T_n). One of the main results
asserts that FID-loci over an acceptable ring are open. We give a module
version of the Nagata criterion, and prove that it holds for all of the
aforementioned properties.
