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.

gollark: This way it wouldn't affect GalactiCRAFT™™™.
gollark: ```category_fallupoutofworld { # Players above this height will get a freezing effect applied to them that slowly obscures their view D:freezingHeight=2000.0 # Players above this height will slowly lose air and take damage if they run out, similar to drowning D:noAirHeight=2500.0 # Players above this height will take rapid damage similar to falling out of the bottom of the world. D:bloodBoilHeight=3000.0 # Damage taken each tick while blood is boiling. 4 would be the same as falling out of the bottom of the world D:bloodBoilDamage=4.0 # Players above this height instantly die D:instantDeathHeight=4000.0}```in `UpAndDownAndAllAround.cfg`.<@!202992030685724675>
gollark: Oh, and I actually was in creative when I died...
gollark: Possibly, but I want to be able to continue using rockets in survival.
gollark: Otherwise your blood will boil as you go up too far.

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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