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

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

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