# 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 $f^{\#}:{\mathcal {O}}_{X}\rightarrow f_{\ast }{\mathcal {O}}_{Z}$ is surjective.
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 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
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.
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 ${\mathcal {I}}\subset {\mathcal {O}}_{X}$ 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 ${\mathcal {I}}{\mathcal {G}}=0$.
A flat closed immersion of finite presentation is the open immersion of an open closed subscheme.