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. The latter condition can be formalized by saying that $f^{#} : 𝒪_{X} \to f_{*} 𝒪_{Z}$ is surjective.

An example is the inclusion map $Spec (R / I) \to Spec (R)$ of affine schemes induced by the canonical ring map $R \to R / I$ .

Other characterizations

The following are equivalent:

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

Definition for locally ringed spaces

In the case of locally ringed spaces 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 $𝒪_{X} \to i_{*} 𝒪_{Z}$ is surjective with kernel $ℐ$
The kernel $ℐ$ is locally generated by sections as an $𝒪_{X}$ -module.

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, $i : 𝔾_{m} ↪ 𝔸^{1}$ where

$𝔾_{m} = Spec (ℤ [x, x^{- 1}])$

If we look at the stalk of $i_{*} 𝒪_{𝔾_{m}} |_{0}$ at $0 \in 𝔸^{1}$ then there are no sections. This implies for any open subscheme $U \subset 𝔸^{1}$ containing $0$ the sheaf has no sections. This violates the third condition since at least one open subscheme $U$ covering $𝔸^{1}$ 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 $X = ⋃ U_{j}$ the induced map $f : f^{- 1} (U_{j}) \to U_{j}$ is a closed immersion.

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.

If $i : Z \to X$ is a closed immersion and $ℐ \subset 𝒪_{X}$ 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 $𝒢$ such that $ℐ 𝒢 = 0$ .

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

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
↑ "Part 4: Algebraic Spaces, Chapter 67: Morphisms of Algebraic Spaces", The stacks project, Columbia University, retrieved 2024-03-06
↑ 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

[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] "Part 4: Algebraic Spaces, Chapter 67: Morphisms of Algebraic Spaces", The stacks project, Columbia University, retrieved 2024-03-06

[7] Grothendieck & Dieudonné 1960, 5.4.6

[8] Stacks, Morphisms of schemes. Lemma 4.1

[9] Stacks, Morphisms of schemes. Lemma 27.2