---
date: 2022-02-23 18:45
modification date: Wednesday 23rd February 2022 18:45:08
title: "closed immersion"
aliases: [closed immersion, closed subscheme]

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


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- Tags 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [scheme](Unsorted/scheme.md)
	- [immersion (topological)](immersion%20(topological))
	- [ideal sheaf](ideal%20sheaf.md)

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# closed immersion

- A **closed immersion** is a morphism $f\in \Sch(Y, X)$  such that 
	- The induced map $\tilde f \in \Top( \abs{X}, \abs{Y})$ is a homeomorphism onto a closed subset of $\abs{Y}$,
	- The induced map $f^* \in \Sh_X(\OO_X, f_* \OO_Y)$ 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.
  
![](attachments/Pasted%20image%2020220418115640.png)

For $\phi \in \CRing(S, R)$ inducing ${}^t\phi \in \Var(\mspec R, \mspec S)$,
![](attachments/Pasted%20image%2020221003231813.png)

# 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
![](attachments/Pasted%20image%2020220405135515.png)

![](attachments/Pasted%20image%2020220405135436.png)