Closed immersion

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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 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]

A basic 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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