Closed immersion

In algebraic geometry, a closed immersion of schemes is a morphism of schemes ${\displaystyle f:Z\to X}$ that identifies Z as a closed subset of X such that locally, regular functions on Z can be extended to X.[1] The latter condition can be formalized by saying that ${\displaystyle f^{\#}:{\mathcal {O}}_{X}\rightarrow f_{\ast }{\mathcal {O}}_{Z}}$ is surjective.[2]

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

An example is the inclusion map ${\displaystyle \operatorname {Spec} (R/I)\to \operatorname {Spec} (R)}$ induced by the canonical map ${\displaystyle R\to R/I}$.

Other characterizations

The following are equivalent:

1. ${\displaystyle f:Z\to X}$ is a closed immersion.
2. For every open affine ${\displaystyle U=\operatorname {Spec} (R)\subset X}$, there exists an ideal ${\displaystyle I\subset R}$ such that ${\displaystyle f^{-1}(U)=\operatorname {Spec} (R/I)}$ as schemes over U.
3. There exists an open affine covering ${\displaystyle X=\bigcup U_{j},U_{j}=\operatorname {Spec} R_{j}}$ and for each j there exists an ideal ${\displaystyle I_{j}\subset R_{j}}$ such that ${\displaystyle f^{-1}(U_{j})=\operatorname {Spec} (R_{j}/I_{j})}$ as schemes over ${\displaystyle U_{j}}$.
4. There is a quasi-coherent sheaf of ideals ${\displaystyle {\mathcal {I}}}$ on X such that ${\displaystyle f_{\ast }{\mathcal {O}}_{Z}\cong {\mathcal {O}}_{X}/{\mathcal {I}}}$ and f is an isomorphism of Z onto the global Spec of ${\displaystyle {\mathcal {O}}_{X}/{\mathcal {I}}}$ 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 ${\displaystyle X=\bigcup U_{j}}$ the induced map ${\displaystyle f:f^{-1}(U_{j})\rightarrow U_{j}}$ is a closed immersion.[3][4]

If the composition ${\displaystyle Z\to Y\to X}$ is a closed immersion and ${\displaystyle Y\to X}$ is separated, then ${\displaystyle Z\to Y}$ is a closed immersion. If X is a separated S-scheme, then every S-section of X is a closed immersion.[5]

If ${\displaystyle i:Z\to X}$ is a closed immersion and ${\displaystyle {\mathcal {I}}\subset {\mathcal {O}}_{X}}$ is the quasi-coherent sheaf of ideals cutting out Z, then the direct image ${\displaystyle i_{*}}$ 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 ${\displaystyle {\mathcal {G}}}$ such that ${\displaystyle {\mathcal {I}}{\mathcal {G}}=0}$.[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 harvnb error: no target: CITEREFHartshorne (help)
3. EGA I, 4.2.4 harvnb error: no target: CITEREFEGA_I (help)
4. http://stacks.math.columbia.edu/download/spaces-morphisms.pdf
5. EGA I, 5.4.6 harvnb error: no target: CITEREFEGA_I (help)
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