# Closed immersion

In algebraic geometry, a **closed immersion** of schemes is a morphism of schemes that identifies *Z* as a closed subset of *X* such that locally, regular functions on *Z* can be extended to *X*.[1] The latter condition can be formalized by saying that is surjective.[2]

*For the same-name concept in differential geometry, see immersion (mathematics).*

An example is the inclusion map induced by the canonical map .

## Other characterizations

The following are equivalent:

- is a closed immersion.
- For every open affine , there exists an ideal such that as schemes over
*U*. - There exists an open affine covering and for each
*j*there exists an ideal such that as schemes over . - There is a quasi-coherent sheaf of ideals on
*X*such that and*f*is an isomorphism of*Z*onto the global Spec of over*X*.

## Properties

A closed immersion is finite and radicial (universally injective). In particular, a closed immersion is universally closed. A closed immersion is stable under base change and composition. The notion of a closed immersion is local in the sense that *f* is a closed immersion if and only if for some (equivalently every) open covering the induced map is a closed immersion.[3][4]

If the composition is a closed immersion and is separated, then is a closed immersion. If *X* is a separated *S*-scheme, then every *S*-section of *X* is a closed immersion.[5]

If is a closed immersion and is the quasi-coherent sheaf of ideals cutting out *Z*, then the direct image from the category of quasi-coherent sheaves over *Z* to the category of quasi-coherent sheaves over *X* is exact, fully faithful with the essential image consisting of such that .[6]

A flat closed immersion of finite presentation is the open immersion of an open closed subscheme.[7]

## See also

## Notes

- Mumford,
*The Red Book of Varieties and Schemes*, Section II.5 - Hartshorne
- EGA I, 4.2.4
- http://stacks.math.columbia.edu/download/spaces-morphisms.pdf
- EGA I, 5.4.6
- Stacks, Morphisms of schemes. Lemma 4.1
- Stacks, Morphisms of schemes. Lemma 27.2

## References

- Grothendieck, Alexandre; Dieudonné, Jean (1960). "Éléments de géométrie algébrique: I. Le langage des schémas".
*Publications Mathématiques de l'IHÉS*.**4**. doi:10.1007/bf02684778. MR 0217083. - The Stacks Project
- Hartshorne, Robin (1977),
*Algebraic Geometry*, Graduate Texts in Mathematics,**52**, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157