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In mathematics, specifically algebraic geometry, the valuative criteria are a collection of results that make it possible to decide 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 : is injective (resp. surjective, resp. bijective). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Valuative criterion」の詳細全文を読む スポンサード リンク
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