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.^[1] The latter condition can be formalized by saying that $f^\#:\mathcal{O}_X\rightarrow f_\ast\mathcal{O}_Z$ is surjective.^[2]

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.^[3][4]

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.^[5]

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 $\mathcal{I} \mathcal{G} = 0$.^[6]

A flat closed immersion of finite presentation is the open immersion of an open closed subscheme.^[7]

See also

• Segre embedding
• Regular embedding

Notes

1. Mumford, The Red Book of Varieties and Schemes, Section II.5
2. Hartshorne
3. EGA I, 4.2.4
4. http://stacks.math.columbia.edu/download/spaces-morphisms.pdf
5. EGA I, 5.4.6
6. Stacks, Morphisms of schemes. Lemma 4.1
7. Stacks, Morphisms of schemes. Lemma 27.2

References

• Grothendieck, Alexandre; Dieudonné, Jean (1960). "Éléments de géométrie algébrique: I. Le langage des schémas" Check |url= value (help). Publications Mathématiques de l'IHÉS 4. doi:10.1007/bf02684778. MR 0217083.
• The Stacks Project
• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157