locally ringed space

locally ringed space

1

Definitions

A

locally ringed space

is a

topological space

X

together with a sheaf of rings

𝒪

X

with the property that, for every point

p

∈

X

, the stalk

(

𝒪

X

)

p

is a

local ring

1

1

All rings mentioned in this article are required to be

commutative

.

.

A morphism of locally ringed spaces from

(

X

,

𝒪

X

)

to

(

Y

,

𝒪

Y

)

is a

continuous map

f

:

X

⟶

Y

together with a

morphism of sheaves

ϕ

:

𝒪

Y

⟶

𝒪

X

with respect to

f

such that, for every point

p

∈

X

, the induced

ring homomorphism

on stalks

ϕ

p

:

(

𝒪

Y

)

f

⁢

(

p

)

⟶

(

𝒪

X

)

p

is a local

homomorphism

. That is,

ϕ

p

⁢

(

y

)

∈

𝔪

p

⁢

for every

⁢

y

∈

𝔪

f

⁢

(

p

)

,

where

𝔪

p

(respectively,

𝔪

f

⁢

(

p

)

) is the

maximal ideal

of the ring

(

𝒪

X

)

p

(respectively,

(

𝒪

Y

)

f

⁢

(

p

)

).

2

Applications

Locally ringed spaces are encountered in many natural

contexts

. Basically, every sheaf on the topological space

X

consisting of

continuous functions

with values in some field is a locally ringed space. Indeed, any such

function

which is not zero at a point

p

∈

X

is nonzero and thus

invertible

in some

neighborhood

of

p

, which implies that the only maximal ideal of the stalk at

p

is the set of germs of functions which vanish at

p

. The utility of this definition lies in the fact that one can then form constructions in familiar

instances

of locally ringed spaces which readily generalize in ways that would not necessarily be obvious without this framework. For example, given a manifold

X

and its locally ringed space

𝒟

X

of real–valued

differentiable functions

, one can show that the space of all

tangent vectors

to

X

at

p

is naturally

isomorphic

to the real vector space

(

𝔪

p

/

𝔪

p

2

)

*

, where the

*

indicates the dual vector space. We then see that, in general, for

any

locally ringed space

X

, the space of tangent vectors at

p

should be defined as the

k

–

vector space

(

𝔪

p

/

𝔪

p

2

)

*

, where

k

is the

residue field

(

𝒪

X

)

p

/

𝔪

p

and

*

denotes dual with respect to

k

as before. It turns out that this definition is the correct definition even in esoteric contexts like

algebraic geometry

over finite fields which at first sight lack the differential structure needed for constructions such as tangent vector.

Another useful application of locally ringed spaces is in the construction of schemes. The

forgetful functor

assigning to each locally ringed space

(

X

,

𝒪

X

)

the ring

𝒪

X

⁢

(

X

)

is

adjoint

to the “

prime spectrum

”

functor

taking each ring

R

to the locally ringed space

Spec

⁡

(

R

)

, and this correspondence is essentially why the

category

of locally ringed spaces is the proper building block to use in the formulation of the notion of scheme.

Title

locally ringed space

Canonical name

LocallyRingedSpace

Date of creation

2013-03-22 12:37:41

Last modified on

2013-03-22 12:37:41

Owner

djao (24)

Last modified by

djao (24)

Numerical id

13

Author

djao (24)

Entry type

Definition

Classification

msc 14A15

Classification

msc 18F20

Related topic

LocalRing

Related topic

PrimeSpectrum

Related topic

Scheme

Defines

morphism of locally ringed spaces

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