Closed immersion

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]

For the same-name concept in differential geometry, see immersion (mathematics).

An example is the inclusion map induced by the canonical map .

Other characterizations

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.


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.[3][4]

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.[5]

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 .[6]

A flat closed immersion of finite presentation is the open immersion of an open closed subscheme.[7]

See also


  1. Mumford, The Red Book of Varieties and Schemes, Section II.5
  2. Hartshorne
  3. EGA I, 4.2.4
  5. EGA I, 5.4.6
  6. Stacks, Morphisms of schemes. Lemma 4.1
  7. 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
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