---
tag: GFC5YKSN5H
kind: concept
name: Open immersion
msc: [14A15]
specializes: [JXRKH6P1Z4]
sources:
  - url: https://en.wikipedia.org/wiki/Glossary_of_algebraic_geometry
---

## Summary

An open immersion is a [[morphism of schemes|JXRKH6P1Z4]] that identifies its source with an open subscheme of its target.
Open immersions are the maps used to glue schemes and to restrict a scheme to an open subset.
The basic example is the inclusion of a basic open set $D(f) = \operatorname{Spec} A_f$ into $\operatorname{Spec} A$.

## Explanations

### L1 human

An open immersion is the inclusion of an open part of a scheme, carrying the same functions as the big scheme there.
Removing the origin from the line gives an open immersion of $\operatorname{Spec} k[x, x^{-1}]$ into $\operatorname{Spec} k[x]$.
On the open part, the function $1/x$ is allowed, because $x$ is never zero there.
Each chart of projective space is an open immersion of affine space into it.