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.
 * 1) 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.
 * 2) 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 $$U_j$$.
 * 3) 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


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