Valuative criterion

In mathematics, specifically algebraic geometry, the valuative criteria is a name for a collection of results that make it possible 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:
#"f" is separated (resp. universally closed, resp. proper)
#"f" is quasi-separated (resp. quasi-compact and separated, resp. of finite type) 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:$ext\{Hom\}\_Y(Y\text{'},\; X)\; o\; ext\{Hom\}\_Y(\; ext\{Spec\}\; K,\; X)$is injective (resp. surjective, resp. bijective).

References

*cite journal
last = Grothendieck
first = Alexandre
authorlink = Alexandre Grothendieck
coauthors = Jean Dieudonné
year = 1961
title = Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : II. Étude globale élémentaire de quelques classes de morphismes | journal = Publications Mathématiques de l'IHÉS
volume = 8
pages = 5–222
url = http://www.numdam.org:80/numdam-bin/feuilleter?id=PMIHES_1961__8_