Valuative Criterion
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High Quality Content by WIKIPEDIA articles! 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. 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. f is separated (resp. universally closed, resp. proper) 2. f is quasi-separat...
High Quality Content by WIKIPEDIA articles! 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. 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. f is separated (resp. universally closed, resp. proper) 2. 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.
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