In algebraic geometry, a closed immersion of schemess 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. is surjective.

of affine schemes induced by the canonical ring map.

Other characterizations

The following are equivalent:

1. is a closed immersion.

2. as schemes over U.

3. and for each j.

4. There is a quasi-coherent sheaf of ideals on X and f is an isomorphism of Z onto the global Spec over X.

Definition for locally ringed spaces

In the case of locally ringed spaces is a closed immersion if a similar list of criteria is satisfied:

1. is a homeomorphism onto its image

2. -module.

where.

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.

is separated is a closed immersion. If X is a separated S-scheme, then every S-section of X is a closed immersion.

is the quasi-coherent sheaf of ideals cutting out Z from the category of quasi-coherent sheaves over Z to the category of quasi-coherent sheaves over X.

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

See also

• Segre embedding

• Regular embedding

Notes

References

• The Stacks Project