Closed immersion

In algebraic geometry, a closed immersion of schemes is a morphism of schemes

f

:

Z

→

X

{\displaystyle 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

f

#

:

O

X

→

f

∗

O

Z

{\displaystyle f^{\#}:{\mathcal {O}}_{X}\rightarrow f_{\ast }{\mathcal {O}}_{Z}}

is surjective.

An example is the inclusion map

Spec

⁡

(

R

/

I

)

→

Spec

⁡

(

R

)

{\displaystyle \operatorname {Spec} (R/I)\to \operatorname {Spec} (R)}

of affine schemes induced by the canonical ring map

R

→

R

/

I

{\displaystyle R\to R/I}

.

Other characterizations

The following are equivalent:

f

:

Z

→

X

{\displaystyle f:Z\to X}

is a closed immersion.

For every open affine

U

=

Spec

⁡

(

R

)

⊂

X

{\displaystyle U=\operatorname {Spec} (R)\subset X}

, there exists an ideal

I

⊂

R

{\displaystyle I\subset R}

such that

f

−

1

(

U

)

=

Spec

⁡

(

R

/

I

)

{\displaystyle f^{-1}(U)=\operatorname {Spec} (R/I)}

as schemes over U.

There exists an open affine covering

X

=

⋃

U

j

,

U

j

=

Spec

⁡

R

j

{\displaystyle X=\bigcup U_{j},U_{j}=\operatorname {Spec} R_{j}}

and for each j there exists an ideal

I

j

⊂

R

j

{\displaystyle I_{j}\subset R_{j}}

such that

f

−

1

(

U

j

)

=

Spec

⁡

(

R

j

/

I

j

)

{\displaystyle f^{-1}(U_{j})=\operatorname {Spec} (R_{j}/I_{j})}

as schemes over

U

j

{\displaystyle U_{j}}

.

There is a quasi-coherent sheaf of ideals

I

{\displaystyle {\mathcal {I}}}

on X such that

f

∗

O

Z

≅

O

X

/

I

{\displaystyle f_{\ast }{\mathcal {O}}_{Z}\cong {\mathcal {O}}_{X}/{\mathcal {I}}}

and f is an isomorphism of Z onto the global Spec of

O

X

/

I

{\displaystyle {\mathcal {O}}_{X}/{\mathcal {I}}}

over X.

Definition for locally ringed spaces

In the case of locally ringed spaces a morphism

i

:

Z

→

X

{\displaystyle i:Z\to X}

is a closed immersion if a similar list of criteria is satisfied:

The map

i

{\displaystyle i}

is a homeomorphism of

Z

{\displaystyle Z}

onto its image

The associated sheaf map

O

X

→

i

∗

O

Z

{\displaystyle {\mathcal {O}}_{X}\to i_{*}{\mathcal {O}}_{Z}}

is surjective with kernel

I

{\displaystyle {\mathcal {I}}}

The kernel

I

{\displaystyle {\mathcal {I}}}

is locally generated by sections as an

O

X

{\displaystyle {\mathcal {O}}_{X}}

-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,

i

:

G

m

↪

A

1

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

where

G

m

=

Spec

(

Z

[

x

,

x

−

1

]

)

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

If we look at the stalk of

i

∗

O

G

m

|

0

{\displaystyle i_{*}{\mathcal {O}}_{\mathbb {G} _{m}}|_{0}}

at

0

∈

A

1

{\displaystyle 0\in \mathbb {A} ^{1}}

then there are no sections. This implies for any open subscheme

U

⊂

A

1

{\displaystyle U\subset \mathbb {A} ^{1}}

containing

0

{\displaystyle 0}

the sheaf has no sections. This violates the third condition since at least one open subscheme

U

{\displaystyle U}

covering

A

1

{\displaystyle \mathbb {A} ^{1}}

contains

0

{\displaystyle 0}

.

Properties

A closed immersion is finite and radicial (universally injective).