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 $\mathrm{Spec}(R/I)\to \mathrm{Spec}(R)$ of affine schemes induced by the canonical ring 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 ${𝒪}_X\to i_{\ast }{𝒪}_Z$ 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, $i:{𝔾}_m\hookrightarrow {𝔸}^1$ where

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

If we look at the stalk of $i_{\ast }{𝒪}_{{𝔾}_m}{|}_0$ 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 $X=\bigcup U_j$ 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 $i_{\ast }$ 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 $ℐ𝒢=0$.

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
• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157