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valuative criterion of properness

The valuative criterion of properness EGA II, 7.3.8 (numdam). It has been found by Chevalley.

Théorème (7.3.8). — Soient Y un préschéma (resp. un préschéma localement noethérien), XYX\to Y un morphisme quasi-compact séparé (resp. de type fini). Les conditions suivantes sont équivalentes :

a) f est universellement fermé (resp. propre).

b) Pour tout YY-schéma de la forme Y=Spec(A)Y'=Spec(A), où AA est un anneau de valuation (resp. anneau de valuation discrète) de corps des fractions KK, l’application canonique

Hom Y(Y,X)Hom Y(Spec(K),X) Hom_Y(Y', X) \to Hom_Y(Spec(K), X)

correspondant à l’injection canonique AKA\to K, est surjective (resp. bijective).

In other words, the class of proper morphisms satisfies the unique right lifting property with respect to the class of morphisms SpecKSpecRSpec K\to Spec R formally dual to the injections of valuation rings into their fields of fractions.

There is a pattern-terminological reinterpretation in Kontsevich-Rosenberg, Noncommutative spaces, 6.8.1:

Let M vM_v' be the family of canonical injections of valuation rings to their fields of fractions, and let M vM_v be its image in the category of functors CRingSetCRing\to Set. Consider schemes as covariant presheaves on CRingCRing. Then a morphism f:XYf:X\to Y of schemes is separated iff it is formally M vM_v-unramified, universally closed iff it is formally M vM_v-smooth and proper iff it is formally M vM_v-étale (in the sense of KR 6.3).

Revised on April 30, 2011 16:27:05 by Zoran Škoda (31.45.185.22)