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 locally, regular functions on Z can be extended to X. The latter condition can be formalized by saying that is surjective. An example is the inclusion map induced by the canonical map R \to R/I.

Other characterizations

The following are equivalent:

Definition for locally ringed spaces

In the case of locally ringed spaces a morphism i:Z\to X is a closed immersion if a similar list of criteria is satisfied 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 0 the sheaf has no sections. This violates the third condition since at least one open subscheme U covering contains 0.

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 the induced map is a closed immersion. If the composition 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. If i: Z \to X is a closed immersion and 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. A flat closed immersion of finite presentation is the open immersion of an open closed subscheme.