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  • 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