# Valuative criterion

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In [mathematics](/source/Mathematics), specifically [algebraic geometry](/source/Algebraic_geometry), the **valuative criteria** are a collection of results that make it possible to decide whether a [morphism](/source/Morphism) of [algebraic varieties](/source/Algebraic_variety), or more generally [schemes](/source/Scheme_(mathematics)), is [universally closed](/source/Universally_closed), [separated](/source/Glossary_of_algebraic_geometry#separated), or [proper](/source/Proper_morphism).

## Statement of the valuative criteria

Recall that a [valuation ring](/source/Valuation_ring) *A* is a domain, so if *K* is the [field of fractions](/source/Field_of_fractions) of *A*, then Spec *K* is the [generic point](/source/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)
1. *f* is [quasi-separated](/source/Quasi-separated) (resp. [quasi-compact](/source/Quasi-compact_morphism), 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

- \text{Hom}_Y(Y', X) \to \text{Hom}_Y(\operatorname{Spec} K, X)

is injective (resp. surjective, resp. bijective).

Furthermore, in the special case when *Y* is (locally) [Noetherian](/source/Noetherian_scheme), it suffices to check the case that *A* is a [discrete valuation ring](/source/Discrete_valuation_ring).

## References

1. EGA II, proposition 7.2.3 and théorème 7.3.8.

1. Stacks Project, tags 01KA, 01KY, and 0BX4.

- Grothendieck, Alexandre & [Jean Dieudonné](/source/Jean_Dieudonn%C3%A9) (1961). ["Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : II. Étude globale élémentaire de quelques classes de morphismes"](https://web.archive.org/web/20170112024503/http://www.numdam.org/numdam-bin/feuilleter?id=PMIHES_1961__8_). *Publications Mathématiques de l'IHÉS*. **8**: 5–222. [doi:10.1007/bf02699291](https://doi.org/10.1007/bf02699291). Archived from [the original](http://www.numdam.org/numdam-bin/feuilleter?id=PMIHES_1961__8_) on 2017-01-12. Retrieved 2008-08-12.

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