prime spectrum

prime spectrum

1

Spec as a set

Let

R

be any

commutative ring

with

identity

. The

prime
spectrum

Spec

⁡

(

R

)

of

R

is defined to be the set

{

P

⊊

R

∣

P

⁢

is a prime ideal of

⁢

R

}

.

For any subset

A

of

R

, we define the

variety

of

A

to be
the set

V

⁢

(

A

)

:=

{

P

∈

Spec

⁡

(

R

)

∣

A

⊂

P

}

⊂

Spec

⁡

(

R

)

It is enough to restrict attention to subsets of

R

which are ideals,
since, for any subset

A

of

R

, we have

V

⁢

(

A

)

=

V

⁢

(

I

)

where

I

is
the

ideal generated by

A

. In fact, even more is true:

V

⁢

(

I

)

=

V

⁢

(

I

)

where

I

denotes the

radical

of the ideal

I

.

2

Spec as a topological space

We impose a

topology

on

Spec

⁡

(

R

)

by defining the sets

V

⁢

(

A

)

to be
the

collection

of

closed subsets

of

Spec

⁡

(

R

)

(that is, a subset of

Spec

⁡

(

R

)

is open if and only if it equals the

complement

of

V

⁢

(

A

)

for some subset

A

). The

equations

⋂

α

V

⁢

(

I

α

)

=

V

⁢

(

⋃

α

I

α

)

⋃

i

=

1

n

V

⁢

(

I

i

)

=

V

⁢

(

⋂

i

=

1

n

I

i

)

,

for any ideals

I

α

,

I

i

of

R

, establish that this collection does constitute a topology on

Spec

⁡

(

R

)

. This topology is called the

Zariski topology

in
light of its relationship to the Zariski topology on an

algebraic
variety

(see

Section

4

below). Note that a point

P

∈

Spec

⁡

(

R

)

is closed if and only if

P

⊂

R

is a

maximal ideal

.

A

distinguished open set

of

Spec

⁡

(

R

)

is defined to be an

open
set

of the form

Spec

(

R

)

f

:=

{

P

∈

Spec

(

R

)

∣

f

∉

P

}

=

Spec

(

R

)

∖

V

(

{

f

}

)

,

for any

element

f

∈

R

. The collection of distinguished open sets
forms a topological basis for the open sets of

Spec

⁡

(

R

)

. In fact, we
have

Spec

(

R

)

∖

V

(

A

)

=

⋃

f

∈

A

Spec

(

R

)

f

.

The

topological space

Spec

⁡

(

R

)

has the following additional
properties:

•

Spec

⁡

(

R

)

is

compact

(but almost never

Hausdorff

).

•

A subset of

Spec

⁡

(

R

)

is an

irreducible

closed set

if and only if it equals

V

⁢

(

P

)

for some

prime ideal

P

of

R

.

•

For

f

∈

R

, let

R

f

denote the

localization

of

R

at

f

. Then the topological spaces

Spec

(

R

)

f

and

Spec

⁡

(

R

f

)

are
naturally

homeomorphic

, via the correspondence sending a prime ideal
of

R

not containing

f

to the induced prime ideal in

R

f

.

•

For

P

∈

Spec

⁡

(

R

)

, let

R

P

denote the localization of

R

at the prime ideal

P

. Then the topological spaces

V

⁢

(

P

)

⊂

Spec

⁡

(

R

)

and

Spec

⁡

(

R

P

)

are naturally homeomorphic, via the
correspondence sending a prime ideal of

R

contained

in

P

to the
induced prime ideal in

R

P

.

3

Spec as a sheaf

For convenience, we adopt the usual convention of writing

X

for

Spec

⁡

(

R

)

. For any

f

∈

R

and

P

∈

X

f

, let

ι

f

,

P

:

R

f

⟶

R

P

be the natural

inclusion map

. Define a

presheaf

of rings

𝒪

X

on

X

by setting

𝒪

X

⁢

(

U

)

:=

{

(

s

P

)

∈

∏

P

∈

U

R

P

|

U

has an open cover

{

X

f

α

}

with elements

∈

s

α

R

f

α

such that

=

s

P

⁢

ι

f

α

,

P

(

s

α

)

whenever

∈

P

X

f

α

}

,

for each open set

U

⊂

X

. The

restriction

map

res

U

,

V

:

𝒪

X

⁢

(

U

)

⟶

𝒪

X

⁢

(

V

)

is the map induced by the

projection map

∏

P

∈

U

R

P

⟶

∏

P

∈

V

R

P

,

for each open subset

V

⊂

U

. The presheaf

𝒪

X

satisfies

the
following properties:

1.

𝒪

X

is a sheaf.

2.

𝒪

X

⁢

(

X

f

)

=

R

f

for every

f

∈

R

.

3.

The stalk

(

𝒪

X

)

P

is equal to

R

P

for every

P

∈

X

. (In particular,

X

is a

locally ringed space

.)

4.

The restriction sheaf of

𝒪

X

to

X

f

is

isomorphic

as a
sheaf to

𝒪

Spec

⁡

(

R

f

)

.

4

Relationship to algebraic varieties

Spec

⁡

(

R

)

is sometimes called an

affine scheme

because of the
close relationship between

affine varieties

in

𝔸

k

n

and the

Spec

of their corresponding

coordinate rings

. In fact, the
correspondence between the two is an

equivalence of categories

,
although a

complete

statement of this

equivalence

requires the notion
of

morphisms of schemes

and will not be given
here. Nevertheless, we explain what we can of this correspondence
below.

Let

k

be a field and write as usual

𝔸

k

n

for the

vector space

k

n

. Recall that an affine variety

V

in

𝔸

k

n

is the set of
common zeros of some prime ideal

I

⊂

k

⁢

[

X

1

,

…

,

X

n

]

. The coordinate ring of

V

is defined to be
the ring

R

:=

k

⁢

[

X

1

,

…

,

X

n

]

/

I

, and there is an

embedding

i

:

V

↪

Spec

⁡

(

R

)

given by

i

⁢

(

a

1

,

…

,

a

n

)

:=

(

X

1

-

a

1

,

…

,

X

n

-

a

n

)

∈

Spec

⁡

(

R

)

.

The

function

i

is not a homeomorphism, because it is not a

bijection

(its image is contained inside the set of maximal ideals of

R

). However,
the map

i

does define an order preserving bijection between the open
sets of

V

and the open sets of

Spec

⁡

(

R

)

in the Zariski topology.
This

isomorphism

between these two lattices of open sets can be used
to equate the sheaf

Spec

⁡

(

R

)

with the

structure sheaf

of the variety

V

, showing that the two objects are identical in every respect
except for the minor detail of

Spec

⁡

(

R

)

having more points than

V

.

The additional points of

Spec

⁡

(

R

)

are valuable in many situations
and a systematic study of them leads to the general notion of
schemes. As just one example, the classical Bezout’s

theorem

is only
valid for algebraically closed fields, but admits a scheme–theoretic

generalization

which holds over non–algebraically closed fields as
well. We will not attempt to explain the theory of schemes in detail,
instead referring the interested reader to the

references

below.

Remark

. The

spectrum

Spec

⁡

(

R

)

of a ring

R

may be generalized to the case when

R

is not

commutative

, as long as

R

contains the

multiplicative identity

. For a ring

R

with

1

, the

Spec

⁡

(

R

)

, like above, is the set of all proper prime ideals of

R

. This definition is used to develop the

noncommutative

version of Hilbert’s

Nullstellensatz

.

References

1

Robin Hartshorne,

Algebraic Geometry

,
Springer–Verlag New York, Inc., 1977 (GTM

52

).

2

David Mumford,

The Red Book of Varieties and
Schemes, Second Expanded Edition

, Springer–Verlag, 1999 (LNM

1358

).

3

Louis H. Rowen,

Ring Theory, Vol. 1

, Academic Press, New York, 1988.

Title

prime spectrum

Canonical name

PrimeSpectrum

Date of creation

2013-03-22 12:38:07

Last modified on

2013-03-22 12:38:07

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

16

Author

CWoo (3771)

Entry type

Definition

Classification

msc 14A15

Related topic

Scheme

Related topic

Sheaf

Related topic

ZariskiTopology

Related topic

LocallyRingedSpace

Defines

distinguished open set

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