ABSTRACT
Sufficient conditions are given for a (unital) homomorphism f: A → B of (commutative) rings to be a chain morphism, in the sense that a f: Spec(B) → Spec(A) permits the covering of chains of arbitrary cardinality. One such sufficient condition is that f satisfies lying-over, a f be open in the flat (resp., Zariski) topology, and that each reduced fiber of a f be quasilocal (resp., an integral domain). Sufficient conditions are given for f to have the generalized going-down property GGD (that is, “going-down” predicated for chains of arbitrary cardinality). Typical of such sufficient conditions are the following: f is a chain morphism and B is quasilocal treed; f satisfies going-down and either the reduced fibers of a f are integral domains or A is a going-down ring. “Universally going-down” is equivalent to “universally GGD”; in particular, if f is flat, then f satisfies GGD. The universally subtrusive homomorphisms are the same as the universally chain morphisms, and these descend the GGD property.
