Closed immersion

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

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

Other characterizations

The following are equivalent:

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

Definition for locally ringed spaces

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

1. The map ${\displaystyle i}$ is a homeomorphism of ${\displaystyle Z}$ onto its image
2. The associated sheaf map ${\displaystyle {\mathcal {O}}_{X}\to i_{*}{\mathcal {O}}_{Z}}$ is surjective with kernel ${\displaystyle {\mathcal {I}}}$
3. The kernel ${\displaystyle {\mathcal {I}}}$ is locally generated by sections as an ${\displaystyle {\mathcal {O}}_{X}}$-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, ${\displaystyle i:\mathbb {G} _{m}\hookrightarrow \mathbb {A} ^{1}}$ where

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

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

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

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

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

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

See also

• Segre embedding
• Regular embedding

Notes

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