Last modified date: <%+ tp.file.last_modified_date() %>

• Tags
  • #todo/untagged
• Refs:
  • #todo/add-references
• Links:
  • scheme
  • immersion (topological)
  • ideal sheaf

closed immersion

• A closed immersion is a morphism \(f\in {\mathsf{Sch}}(Y, X)\) such that
  • The induced map \(\tilde f \in {\mathsf{Top}}( {\left\lvert {X} \right\rvert}, {\left\lvert {Y} \right\rvert})\) is a homeomorphism onto a closed subset of \({\left\lvert {Y} \right\rvert}\),
  • The induced map \(f^* \in {\mathsf{Sh}}_X({\mathcal{O}}_X, f_* {\mathcal{O}}_Y)\) is a surjection.
• A closed immersion is a morphism \(f: Y \rightarrow X\) of schemes such that \(f\) induces a homeomorphism of the underlying space of \(Y\) onto a closed subset of the underlying space of \(X\) and furthermore the induced map \(f^{\sharp}: \mathscr{O}_{X} \rightarrow f_{*} \mathscr{O}_{Y}\) of sheaves on \(X\) is surjective.

For \(\phi \in \mathsf{CRing}(S, R)\) inducing \({}^t\phi \in {\mathsf{Var}}(\operatorname{mSpec}R, \operatorname{mSpec}S)\),

closed subscheme

A closed subscheme of \(X\) is an equivalence class of closed immersions where \(f: Y \rightarrow X\) and \(f^{\prime}: Y^{\prime} \rightarrow X\) are equivalent if there is an isomorphism \(\varphi: Y^{\prime} \rightarrow Y\) such that \(f^{\prime}=f \circ \varphi\).

Examples