---
tag: WWRWYSJDET
kind: concept
name: Closed immersion
msc: [14A15]
specializes: [JXRKH6P1Z4]
sources:
  - url: https://en.wikipedia.org/wiki/Closed_immersion
same_as:
  wikidata: Q5135342
  wikipedia: Closed immersion
---

## Summary

A closed immersion is a [[morphism of schemes|JXRKH6P1Z4]] that identifies its source with a closed subscheme of its target.
For affine schemes, closed immersions into $\operatorname{Spec} A$ are exactly the maps $\operatorname{Spec} A/I \to \operatorname{Spec} A$ for ideals $I$.
They make precise the idea of a subvariety cut out by equations, and they include non-reduced subschemes such as a double point.

## Explanations

### L1 human

A closed immersion is the scheme version of "the set of solutions of some equations inside a bigger space".
The equations $I$ in $A$ cut out $\operatorname{Spec} A/I$, and every function on it comes from a function on the big space.
The equation $x^2 = 0$ on the line cuts out the origin with a double structure.
This is a different closed subscheme from the origin cut out by $x = 0$, though both have the same single point.