Valuative criterion

In mathematics, specifically algebraic geometry, the valuative criteria are a collection of results that make it possible to decide whether a morphism of algebraic varieties, or more generally schemes, is universally closed, separated, or proper.
Statement of the valuative criteria.
Recall that a valuation ring A is a domain, so if "K" is the field of fractions of "A", then Spec "K" is the generic point of Spec "A".
Let "X" and "Y" be schemes, and let "f" : "X" → "Y" be a morphism of schemes. Then the following are equivalent:
The lifting condition is equivalent to specifying that the natural morphism
is injective (resp. surjective, resp. bijective).
Furthermore, in the special case when "Y" is (locally) noetherian, it suffices to check the case that "A" is a discrete valuation ring.