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.
Read more about Proper Morphism: Definition, Examples, Properties and Characterizations of Proper Morphisms, Proper Morphism of Formal Schemes
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