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- For the same-name concept in differential geometry, see immersion (mathematics).
In algebraic geometry, a closed immersion of schemes is a morphism of schemes 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 is surjective.
An example is the inclusion map induced by the canonical map .
The following are equivalent:
- is a closed immersion.
- For every open affine , there exists an ideal such that as schemes over U.
- There exists an open affine covering and for each j there exists an ideal such that as schemes over .
- There is a quasi-coherent sheaf of ideals on X such that and f is an isomorphism of Z onto the global Spec of over X.
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 the induced map is a closed immersion.
If is a closed immersion and is the quasi-coherent sheaf of ideals cutting out Z, then the direct image 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 such that .
A flat closed immersion of finite presentation is the open immersion of an open closed subscheme.
- Mumford, The Red Book of Varieties and Schemes, Section II.5
- EGA I, 4.2.4
- EGA I, 5.4.6
- Stacks, Morphisms of schemes. Lemma 4.1
- Stacks, Morphisms of schemes. Lemma 27.2
- 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