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 $f^{\mathrm{\#}}:{{𝒪}}_X\to f_{\ast }{{𝒪}}_Z$ is surjective.

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

Other characterizations

The following are equivalent:

1. $f:Z\to X$ is a closed immersion.
2. For every open affine $U=\mathrm{Spec}(R)\subset X$, there exists an ideal $I\subset R$ such that $f^{-1}(U)=\mathrm{Spec}(R/I)$ as schemes over U.
3. There exists an open affine covering $X=\bigcup U_j,U_j=\mathrm{Spec}{R}_j$ and for each j there exists an ideal $I_j\subset R_j$ such that $f^{-1}(U_j)=\mathrm{Spec}(R_j/I_j)$ as schemes over $U_j$.
4. There is a quasi-coherent sheaf of ideals ${ℐ}$ on X such that $f_{\ast }{{𝒪}}_Z\cong {{𝒪}}_X/{ℐ}$ 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

1. The map $i$ is a homeomorphism of $Z$ onto its image
2. The associated sheaf map ${{𝒪}}_X\to i_{*}{{𝒪}}_Z$ is surjective with kernel ${ℐ}$
3. 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,

$i:{\mathbb{𝔾}}_m\hookrightarrow {\mathbb{𝔸}}^1$

where

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

If we look at the stalk of

$i_{*}{{𝒪}}_{{\mathbb{𝔾}}_m}{|}_0$

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 $f:f^{-1}(U_j)\to U_j$ 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.

See also

• Segre embedding
• Regular embedding

References

• Template:EGA
• The Stacks Project
• Template:Hartshorne AG