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