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
[2] An example is the inclusion map
induced by the canonical map
The following are equivalent: In the case of locally ringed spaces[3] a morphism
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,
This implies for any open subscheme
the sheaf has no sections.
This violates the third condition since at least one open subscheme
A closed immersion is finite and radicial (universally injective).
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.
is a closed immersion and
is a closed immersion.
If X is a separated S-scheme, then every S-section of X is a closed immersion.
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
[8] A flat closed immersion of finite presentation is the open immersion of an open closed subscheme.