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]
- f is separated (resp. universally closed, resp. proper)
- 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.
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gollark: Some of it is compiled from the originals of my projects, some isn't.
References
- EGA II, proposition 7.2.3 and théorème 7.3.8.
- 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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