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In algebraic geometry, a proper morphism between schemes is a scheme-theoretic analogue of a proper map between complex-analytic varieties. A basic example is a complete variety (e.g., projective variety) in the following sense: a ''k''-variety ''X'' is complete in the classical definition if it is universally closed. A proper morphism is a generalization of this to schemes. A closed immersion is proper. A morphism is finite if and only if it is proper and quasi-finite. == Definition == A morphism ''f'' : ''X'' → ''Y'' of algebraic varieties or more generally of schemes, is called universally closed if for all morphisms ''Z'' → ''Y'', the projections for the fiber product : are closed maps of the underlying topological spaces. A morphism ''f'' : ''X'' → ''Y'' of algebraic varieties is called proper if it is separated and universally closed. A morphism of schemes is called proper if it is separated, of finite type and universally closed (() II, 5.4.1 ()). One also says that ''X'' is proper over ''Y''. A variety ''X'' over a field ''k'' is complete when the structural morphism from ''X'' to the spectrum of ''k'' is proper. 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「proper morphism」の詳細全文を読む スポンサード リンク
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