Closed immersion

Other characterizations

The following are equivalent:

1. f : Z → X is a closed immersion.
2. For every open affine U = Spec ⁡ ( R ) ⊂ X , there exists an ideal I ⊂ R such that f − 1 ( U ) = Spec ⁡ ( R / I ) as schemes over U.
3. There exists an open affine covering X = ⋃ U j , U j = Spec ⁡ R j and for each j there exists an ideal I j ⊂ R j such that f − 1 ( U j ) = Spec ⁡ ( R j / I j ) as schemes over U j .
4. There is a quasi-coherent sheaf of ideals I on X such that f ∗ O Z ≅ O X / I and f is an isomorphism of Z onto the global Spec of O X / I over X.

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 X = ⋃ U j the induced map f : f − 1 ( U j ) → U j is a closed immersion.

If the composition Z → Y → X is a closed immersion and Y → X is separated, then Z → 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 → X is a closed immersion and I ⊂ O X 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 G such that I G = 0 .

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