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closed immersion (Definition)

A morphism of schemes $ f: (X,\O _X) \longrightarrow (Y,\O _Y)$ is a closed immersion if:

  1. As a map of topological spaces, $ f: X \longrightarrow Y$ is a homeomorphism from $ X$ into a closed subset of $ Y$, and
  2. the morphism of sheaves $ \O _Y \longrightarrow \O _X$ associated with $ f$ is an epimorphism in the category of sheaves.

This notion is the analog of the notion of closed immersion in the category of differential manifolds.



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Cross-references: differential manifolds, sheaves, category, morphism of sheaves, closed subset, homeomorphism, topological spaces, map, morphism of schemes
There are 2 references to this entry.

This is version 3 of closed immersion, born on 2002-07-14, modified 2004-03-30.
Object id is 3165, canonical name is ClosedImmersion.
Accessed 2661 times total.

Classification:
AMS MSC14A15 (Algebraic geometry :: Foundations :: Schemes and morphisms)

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