Closed immersion explained

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.^[1] The latter condition can be formalized by saying that f ^{\#: l{O}} _{X →} f _{\astl{O}} _{Z} is surjective.

An example is the inclusion map

\operatorname{Spec}(R/I) \to \operatorname{Spec}(R) induced by the canonical map R \to R/I .

Other characterizations

The following are equivalent:

f: Z \to X is a closed immersion.

1. For every open affine

U = \operatorname{Spec}(R) \subset X , there exists an ideal I \subset R such that f ^{-1} (U) = \operatorname{Spec}(R/I) as schemes over U.

1. There exists an open affine covering

X = c up U _{j,} U _{j} = \operatorname{Spec} R _{j} and for each j there exists an ideal I _{j} \subset R _{j} such that f ^{-1} (U _{j)} = \operatorname{Spec} (R _{j} / I _{j)} as schemes over U _{j} .

1. There is a quasi-coherent sheaf of ideals

l{I} on X such that f _{\astl{O}} _{Z\cong} l{O} _{X/l{I}} and f is an isomorphism of Z onto the global Spec of l{O} _{X/l{I}} over X.

Definition for locally ringed spaces

In the case of locally ringed spaces^[2] a morphism

i:Z\to X is a closed immersion if a similar list of criteria is satisfied

1. The map

i is a homeomorphism of Z onto its image

1. The associated sheaf map

l{O} _{X} \to i _{*l{O}} _{Z} is surjective with kernel l{I}

1. The kernel

l{I} is locally generated by sections as an l{O} _{X} -module^[3]

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,

i: G _{m\hookrightarrow} A ^{1} where

G _{m} = Spec ( Z [x,x ^{-1} ])

If we look at the stalk of i _{*l{O}} ^{G _{m}} | _{0} at 0 \in A ^{1} then there are no sections. This implies for any open subscheme U \subset A ^{1} containing 0 the sheaf has no sections. This violates the third condition since at least one open subscheme U covering A ^{1} 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

X = c up U _{j} the induced map f:f ^{-1} (U _{j) →} U _{j} is a closed immersion.

If the composition

Z \to Y \to X 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 l{I} \subset l{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 l{G} such that l{I} l{G} = 0 .^[4]

A flat closed immersion of finite presentation is the open immersion of an open closed subscheme.^[5]

See also

• Segre embedding
• Regular embedding

References

• • The Stacks Project

Notes and References

1. Mumford, The Red Book of Varieties and Schemes, Section II.5
2. Web site: Section 26.4 (01HJ): Closed immersions of locally ringed spaces—The Stacks project. 2021-08-05. stacks.math.columbia.edu.
3. Web site: Section 17.8 (01B1): Modules locally generated by sections—The Stacks project. 2021-08-05. stacks.math.columbia.edu.
4. Stacks, Morphisms of schemes. Lemma 4.1
5. Stacks, Morphisms of schemes. Lemma 27.2