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Concept
Closed immersion
Summary
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. The latter condition can be formalized by saying that is surjective.
An example is the inclusion map induced by the canonical map .
The following are equivalent:
is a closed immersion.
For every open affine , there exists an ideal such that as schemes over U.
There exists an open affine covering and for each j there exists an ideal such that as schemes over .
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.
In the case of locally ringed spaces a morphism is a closed immersion if a similar list of criteria is satisfied
The map is a homeomorphism of onto its image
The associated sheaf map is surjective with kernel
The kernel is locally generated by sections as an -module
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, whereIf we look at the stalk of at then there are no sections. This implies for any open subscheme containing the sheaf has no sections. This violates the third condition since at least one open subscheme covering contains .
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 is separated, then is a closed immersion. If X is a separated S-scheme, then every S-section of X is a closed immersion.
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 .
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