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". 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