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][2]

  1. f is separated (resp. universally closed, resp. proper)
  2. f is quasi-separated (resp. quasi-compact, resp. of finite type and 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 (resp. at least one, resp. exactly one) lift Y' X.

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.

References

  1. EGA II, proposition 7.2.3 and théorème 7.3.8.
  2. Stacks Project, tags 01KA, 01KY, and 0BX4.
  • Grothendieck, Alexandre; Jean Dieudonné (1961). "Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : II. Étude globale élémentaire de quelques classes de morphismes". Publications Mathématiques de l'IHÉS. 8: 5–222. doi:10.1007/bf02699291.


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