Closed immersion

For the same-name concept in differential geometry, see immersion (mathematics).

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 regular functions on Z can be extended locally 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:

1. $f: Z \to X$ is a closed immersion.
2. 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.
3. 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 $U_j$.
4. 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.

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.

See also

• Segre embedding
• Regular embedding

References

• The Stacks Project