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 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 is surjective.[2]

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

Other characterizations[edit]

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.

Properties[edit]

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[edit]

Notes[edit]

  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[edit]