In algebraic geometry, a closed immersion of schemes is a morphism of schemes 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 is surjective.[2]
An example is the inclusion map induced by the canonical map .
Other characterizations
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.
Definition for locally ringed spaces
In the case of locally ringed spaces[3] a morphism is a closed immersion if a similar list of criteria is satisfied
- The map is a homeomorphism of onto its image
- The associated sheaf map is surjective with kernel
- The kernel is locally generated by sections as an -module[4]
The only varying condition is the third. It is instructive to look at a counter-example to get a feel for what the third condition yields by looking at a map which is not a closed immersion, where
If we look at the stalk of at then there are no sections. This implies for any open subscheme containing the sheaf has no sections. This violates the third condition since at least one open subscheme covering contains .
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 the induced map is a closed immersion.[5][6]
If the composition is a closed immersion and is separated, then is a closed immersion. If X is a separated S-scheme, then every S-section of X is a closed immersion.[7]
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 .[8]
A flat closed immersion of finite presentation is the open immersion of an open closed subscheme.[9]
See also
Notes
- ^ Mumford, The Red Book of Varieties and Schemes, Section II.5
- ^ Hartshorne 1977, §II.3
- ^ "Section 26.4 (01HJ): Closed immersions of locally ringed spaces—The Stacks project". stacks.math.columbia.edu. Retrieved 2021-08-05.
- ^ "Section 17.8 (01B1): Modules locally generated by sections—The Stacks project". stacks.math.columbia.edu. Retrieved 2021-08-05.
- ^ Grothendieck & Dieudonné 1960, 4.2.4
- ^ "Part 4: Algebraic Spaces, Chapter 67: Morphisms of Algebraic Spaces", The stacks project, Columbia University, retrieved 2024-03-06
- ^ Grothendieck & Dieudonné 1960, 5.4.6
- ^ Stacks, Morphisms of schemes. Lemma 4.1
- ^ 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, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
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