  We investigate Cohen factorizations of local ring homomorphisms from three
perspectives. First, we prove a "weak functoriality" result for Cohen
factorizations: certain morphisms of local ring homomorphisms induce morphisms
of Cohen factorizations. Second, we use Cohen factorizations to study the
properties of local ring homomorphisms (Gorenstein, Cohen-Macaulay, etc.) in
certain commutative diagrams. Third, we use Cohen factorizations to investigate
the structure of quasi-deformations of local rings, with an eye on the question
of the behavior of CI-dimension in short exact sequences.
