Closed immersion

In algebraic geometry, a closed immersion of schemes is a morphism of schemes $f:Z\to X$ 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 $f^{\#}:{\mathcal {O}}_{X}\rightarrow f_{\ast }{\mathcal {O}}_{Z}$ is surjective.[2]

An example is the inclusion map $\operatorname {Spec} (R/I)\to \operatorname {Spec} (R)$ induced by the canonical map $R\to R/I$ .

Other characterizations

The following are equivalent:

$f:Z\to X$ is a closed immersion.
For every open affine $U=\operatorname {Spec} (R)\subset X$ , there exists an ideal $I\subset R$ such that $f^{-1}(U)=\operatorname {Spec} (R/I)$ as schemes over U.
There exists an open affine covering $X=\bigcup U_{j},U_{j}=\operatorname {Spec} R_{j}$ and for each j there exists an ideal $I_{j}\subset R_{j}$ such that $f^{-1}(U_{j})=\operatorname {Spec} (R_{j}/I_{j})$ as schemes over $U_{j}$ .
There is a quasi-coherent sheaf of ideals ${\mathcal {I}}$ on X such that $f_{\ast }{\mathcal {O}}_{Z}\cong {\mathcal {O}}_{X}/{\mathcal {I}}$ and f is an isomorphism of Z onto the global Spec of ${\mathcal {O}}_{X}/{\mathcal {I}}$ over X.

Definition for locally ringed spaces

In the case of locally ringed spaces[3] a morphism $i:Z\to X$ is a closed immersion if a similar list of criteria is satisfied

The map $i$ is a homeomorphism of $Z$ onto its image
The associated sheaf map ${\mathcal {O}}_{X}\to i_{*}{\mathcal {O}}_{Z}$ is surjective with kernel ${\mathcal {I}}$
The kernel ${\mathcal {I}}$ is locally generated by sections as an ${\mathcal {O}}_{X}$ -module[4]

The only varying condition is the third. It is instructive to look at a counter-example to get a feel for what the third condition yields by looking at a map which is not a closed immersion, $i:\mathbb {G} _{m}\hookrightarrow \mathbb {A} ^{1}$ where

$\mathbb {G} _{m}={\text{Spec}}(\mathbb {Z} [x,x^{-1}])$

If we look at the stalk of $i_{*}{\mathcal {O}}_{\mathbb {G} _{m}}|_{0}$ at $0\in \mathbb {A} ^{1}$ then there are no sections. This implies for any open subscheme $U\subset \mathbb {A} ^{1}$ containing $0$ the sheaf has no sections. This violates the third condition since at least one open subscheme $U$ covering $\mathbb {A} ^{1}$ contains $0$ .

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 $X=\bigcup U_{j}$ the induced map $f:f^{-1}(U_{j})\rightarrow U_{j}$ is a closed immersion.[5][6]

If the composition $Z\to Y\to X$ is a closed immersion and $Y\to X$ is separated, then $Z\to Y$ is a closed immersion. If X is a separated S-scheme, then every S-section of X is a closed immersion.[7]

If $i:Z\to X$ is a closed immersion and ${\mathcal {I}}\subset {\mathcal {O}}_{X}$ is the quasi-coherent sheaf of ideals cutting out Z, then the direct image $i_{*}$ 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 ${\mathcal {G}}$ such that ${\mathcal {I}}{\mathcal {G}}=0$ .[8]

A flat closed immersion of finite presentation is the open immersion of an open closed subscheme.[9]

Notes

Mumford, The Red Book of Varieties and Schemes, Section II.5
Hartshorne 1977, §II.3
"Section 26.4 (01HJ): Closed immersions of locally ringed spaces—The Stacks project". stacks.math.columbia.edu. Retrieved 2021-08-05.
"Section 17.8 (01B1): Modules locally generated by sections—The Stacks project". stacks.math.columbia.edu. Retrieved 2021-08-05.
Grothendieck & Dieudonné 1960, 4.2.4
http://stacks.math.columbia.edu/download/spaces-morphisms.pdf
Grothendieck & Dieudonné 1960, 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, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157

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[1] Mumford, The Red Book of Varieties and Schemes, Section II.5

[2] Hartshorne 1977, §II.3

[3] "Section 26.4 (01HJ): Closed immersions of locally ringed spaces—The Stacks project". stacks.math.columbia.edu. Retrieved 2021-08-05.

[4] "Section 17.8 (01B1): Modules locally generated by sections—The Stacks project". stacks.math.columbia.edu. Retrieved 2021-08-05.

[5] Grothendieck & Dieudonné 1960, 4.2.4

[6] ttp://stacks.math.columbia.edu/download/spaces-morphisms.pdf

[7] Grothendieck & Dieudonné 1960, 5.4.6

[8] Stacks, Morphisms of schemes. Lemma 4.1

[9] Stacks, Morphisms of schemes. Lemma 27.2