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

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The following are equivalent:

  1.   is a closed immersion.
  2. For every open affine  , there exists an ideal   such that   as schemes over U.
  3. There exists an open affine covering   and for each j there exists an ideal   such that   as schemes over  .
  4. 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

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In the case of locally ringed spaces[3] a morphism   is a closed immersion if a similar list of criteria is satisfied

  1. The map   is a homeomorphism of   onto its image
  2. The associated sheaf map   is surjective with kernel  
  3. 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

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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

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Notes

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  1. ^ Mumford, The Red Book of Varieties and Schemes, Section II.5
  2. ^ Hartshorne 1977, §II.3
  3. ^ "Section 26.4 (01HJ): Closed immersions of locally ringed spaces—The Stacks project". stacks.math.columbia.edu. Retrieved 2021-08-05.
  4. ^ "Section 17.8 (01B1): Modules locally generated by sections—The Stacks project". stacks.math.columbia.edu. Retrieved 2021-08-05.
  5. ^ Grothendieck & Dieudonné 1960, 4.2.4
  6. ^ "Part 4: Algebraic Spaces, Chapter 67: Morphisms of Algebraic Spaces", The stacks project, Columbia University, retrieved 2024-03-06
  7. ^ Grothendieck & Dieudonné 1960, 5.4.6
  8. ^ Stacks, Morphisms of schemes. Lemma 4.1
  9. ^ Stacks, Morphisms of schemes. Lemma 27.2

References

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