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:
  1. f is separated
  2. f is quasi-separated and for every valuation ring A, if Y' = Spec A and X' denotes the generic point of Y' , then for every morphism Y' → Y and every morphism X' → X which lifts the generic point, then there exists at most one lift Y' → X.
The lifting condition is equivalent to specifying that the natural morphism
is injective.
Furthermore, in the special case when Y is noetherian, it suffices to check the case that A is a discrete valuation ring.