localization of a module

localization of a module

Let

R

be a

commutative ring

and

M

an

R

-module. Let

S

⊂

R

be a non-empty

multiplicative set

. Form the

Cartesian product

M

×

S

, and define a

binary relation

∼

on

M

×

S

as follows:

(

m

1

,

s

1

)

∼

(

m

2

,

s

2

)

if and only if there is some

t

∈

S

such that

t

⁢

(

s

2

⁢

m

1

-

s

1

⁢

m

2

)

=

0

Proposition 1

.

∼

on

M

×

S

is an

equivalence relation

.

Proof.

Clearly

(

m

,

s

)

∼

(

m

,

s

)

as

t

⁢

(

s

⁢

m

-

s

⁢

m

)

=

0

for any

t

∈

S

, where

S

≠

∅

. Also,

(

m

1

,

s

1

)

∼

(

m

2

,

s

2

)

implies that

(

m

2

,

s

2

)

∼

(

m

1

,

s

1

)

, since

t

⁢

(

s

2

⁢

m

1

-

s

1

⁢

m

2

)

=

0

implies that

t

⁢

(

s

1

⁢

m

2

-

s

2

⁢

m

1

)

=

0

. Finally, given

(

m

1

,

s

1

)

∼

(

m

2

,

s

2

)

and

(

m

2

,

s

2

)

∼

(

m

3

,

s

3

)

, we are led to two

equations

t

⁢

(

s

2

⁢

m

1

-

s

1

⁢

m

2

)

=

0

and

u

⁢

(

s

3

⁢

m

2

-

s

2

⁢

m

3

)

=

0

for some

t

,

u

∈

S

. Expanding and rearranging these, then multiplying the first equation by

u

⁢

s

3

and the second by

t

⁢

s

1

, we get

t

⁢

u

⁢

s

2

⁢

(

s

3

⁢

m

1

-

s

1

⁢

m

3

)

=

0

. Since

t

⁢

u

⁢

s

2

∈

S

,

(

m

1

,

s

1

)

∼

(

m

3

,

s

3

)

as required.
∎

Let

M

S

be the set of

equivalence classes

in

M

×

S

under

∼

. For each

(

m

,

s

)

∈

M

×

S

, write

[

(

m

,

s

)

]

⁢

or more commonly

⁢

m

s

the equivalence class in

M

S

containing

(

m

,

s

)

. Next,

•

define a

binary operation

+

on

M

S

as follows:

m

1

s

1

+

m

2

s

2

:=

s

2

⁢

m

1

+

s

1

⁢

m

2

s

1

⁢

s

2

.

•

define a

function

⋅

:

R

S

×

M

S

→

M

S

as follows:

r

s

⋅

m

t

:=

r

⁢

m

s

⁢

t

where

R

S

is the

localization

of

R

over

S

.

Proposition 2

.

M

S

together with

+

and

⋅

defined above is a

unital module

over

R

S

.

Proof.

That

+

and

⋅

are well-defined is based on the following: if

(

m

1

,

s

1

)

∼

(

m

2

,

s

2

)

, then

m

s

+

m

1

s

1

=

m

s

+

m

2

s

2

,

m

1

s

1

+

m

s

=

m

2

s

2

+

m

s

,

and

r

s

⋅

m

1

s

1

=

r

s

⋅

m

2

s

2

,

which are clear by

Proposition

1

. Furthermore

+

is

commutative

and

associative

and that

⋅

distributes over

+

on both sides, which are all properties inherited from

M

. Next,

0

s

is the

additive identity

in

M

S

and

-

m

s

∈

M

S

is the

additive inverse

of

m

s

. So

M

S

is a module over

R

S

. Finally, since

(

m

⁢

t

,

s

⁢

t

)

∼

(

m

,

s

)

for any

t

∈

S

,

t

t

⋅

m

s

=

m

s

so that

M

S

is unital. ∎

Definition

.

M

S

, as an

R

S

-module, is called the

localization

of

M

at

S

.

M

S

is also written

S

-

1

⁢

M

.

Remarks

.

•

The notion of the localization of a module generalizes that of a ring in the sense that

R

S

is the localization of

R

at

S

as an

R

S

-module.

•

If

S

=

R

-

𝔭

, where

𝔭

is a

prime ideal

in

R

, then

M

S

is usually written

M

𝔭

.

Title

localization of a module

Canonical name

LocalizationOfAModule

Date of creation

2013-03-22 17:26:59

Last modified on

2013-03-22 17:26:59

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 13B30

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