graded module

graded module

It is a well-known fact that a

polynomial

(over, say,

ℤ

) can be written as a sum of

monomials

in a unique way. A monomial is a special kind of a polynomial. Unlike polynomials, the monomials can be partitioned so that the sum of any two monomial within a

partition

, and the

product

of any two monomials, are again monomials. As one may have guessed, one would partition the monomials by their degree. The above notion can be generalized, and the general notion is that of a

graded ring

(and a

graded module

).

Definition

Let

R

=

R

0

⊕

R

1

⊕

⋯

be a graded ring. A module

M

over

R

is said to be a

graded module

if

M

=

M

0

⊕

M

1

⊕

⋯

where

M

i

are

abelian

subgroups

of

M

, such that

R

i

⁢

M

j

⊆

M

i

+

j

for all

i

,

j

. An element of

M

is said to be

homogeneous of degree

i

if it is in

M

i

. The set of

M

i

is called a

grading

of

M

.

Whenever we speak of a graded module, the module is always assumed to be over a graded ring. As any ring

R

is trivially a graded ring (where

R

i

=

R

if

i

=

0

and

R

i

=

0

otherwise), every module

M

is trivially a graded module with

M

i

=

M

if

i

=

0

and

M

i

=

0

otherwise. However, it is customary to regard a graded module (or a graded ring) non-trivially.

If

R

is a graded ring, then clearly it is a graded module over itself, by setting

M

i

=

R

i

(

M

=

R

in this case). Furthermore, if

M

is graded over

R

, then so is

M

⁢

z

for any

indeterminate

z

.

Example

. To see a concrete example of a graded module, let us first

construct

a graded ring. For convenience, take any ring

R

, the

polynomial ring

S

=

R

⁢

[

x

]

is a graded ring, as

S

=

S

0

⊕

S

1

⊕

S

2

⊕

⋯

⊕

S

n

⊕

⋯

,

with

S

i

:=

R

⁢

x

i

. Then

S

i

⁢

S

j

=

(

R

⁢

x

m

)

⁢

(

R

⁢

x

n

)

⊆

R

⁢

x

m

+

n

=

S

i

+

j

.

Therefore,

S

is a graded module over

S

. Similarly, the

submodules

S

⁢

x

i

of

S

are also graded over

S

.

It is possible for a module over a graded ring to be graded in more than one way. Let

S

be defined as in the example above. Then

S

⁢

[

y

]

is graded over

S

. One way to grade

S

⁢

[

y

]

is the following:

S

⁢

[

y

]

=

⊕

k

=

0

∞

A

k

,

where

⁢

A

k

=

R

⁢

[

y

]

⁢

x

k

,

since

S

p

⁢

A

q

=

(

R

⁢

x

p

)

⁢

(

R

⁢

[

y

]

⁢

x

q

)

⊆

R

⁢

[

y

]

⁢

x

p

+

q

=

A

p

+

q

. Another way to grade

S

⁢

[

y

]

is:

S

⁢

[

y

]

=

⊕

k

=

0

∞

B

k

,

where

⁢

B

k

=

∑

i

+

j

=

k

R

⁢

x

i

⁢

y

j

,

since

S

p

⁢

B

q

=

(

R

⁢

x

p

)

⁢

(

∑

i

+

j

=

q

R

⁢

x

i

⁢

y

j

)

=

∑

i

+

j

=

q

R

⁢

x

i

+

p

⁢

y

j

⊆

∑

i

+

j

=

p

+

q

R

⁢

x

i

⁢

y

j

=

B

p

+

q

.

Graded homomorphisms and graded submodules

Let

M

,

N

be graded modules over a (graded) ring

R

. A

module homomorphism

f

:

M

→

N

is said to be

graded

if

f

⁢

(

M

i

)

⊆

N

i

.

f

is a

graded

isomorphism

if it is a graded module homomorphism and an isomorphism. If

f

is a graded isomorphism

M

→

N

, then

1.

f

⁢

(

M

i

)

=

N

i

. Suppose

a

∈

N

i

and

f

⁢

(

b

)

=

a

. Write

b

=

∑

b

j

where

b

j

∈

M

i

, and

b

j

=

0

for all but finitely many

j

. Then each

f

⁢

(

b

j

)

∈

N

j

. Since

N

j

∩

N

i

=

0

if

i

≠

j

,

f

⁢

(

b

j

)

=

0

if

j

≠

i

. Therefore

b

=

b

i

∈

N

i

.

2.

f

-

1

is graded. If

a

∈

f

-

1

⁢

(

N

i

)

, then

f

⁢

(

a

)

∈

N

i

=

f

⁢

(

M

i

)

by the previous fact. Then

f

⁢

(

a

)

=

f

⁢

(

c

)

for some

c

∈

M

i

, so

a

=

c

∈

M

i

since

f

is

one-to-one

.

Suppose a graded module

M

has two gradings:

M

=

⊕

A

i

=

⊕

B

i

. The two gradings on

M

are said to be

isomorphic

if there is a graded isomorphism

f

on

M

with

f

⁢

(

A

j

)

=

B

j

. In the example above, we see that the two gradings of

S

⁢

[

j

]

are non-isomorphic.

Let

N

be a submodule of a graded module

M

(over

R

). We can turn

N

into a graded module by defining

N

i

=

N

∩

M

i

. Of course,

N

may already be a graded module in the first place. But the two gradings on

N

may not be isomorphic. A submodule

N

of a graded module

M

(over

R

) is said to be a

graded submodule

of

M

if its grading is defined by

N

i

=

N

∩

M

i

. If

N

is a graded submodule of

M

, then the injection

N

↦

M

is a graded homomorphism.

Generalization

The above definition can be generalized, and the

generalization

comes from the

subscripts

. The set of subscripts in the definition above is just the set of all non-negative integers (sometimes denoted

ℕ

) with a

binary operation

+

. It is reasonable to extend the set of subscripts from

ℕ

to an arbitrary set

S

with a binary operation

*

. Normally, we require that

*

is

associative

so that

S

is a

semigroup

. An

R

-module

M

is said to be

S

-graded if

M

=

⊕

s

∈

S

M

s

⁢

such that

⁢

R

s

⁢

M

t

⊆

M

s

*

t

.

Examples of

S

-graded modules are mainly found in modules over a semigroup ring

R

⁢

[

S

]

.

Remark

. Graded modules, and in general the

concept

of grading in

algebra

, are an essential tool in the study of homological

algebraic

aspect of rings.

Title

graded module

Canonical name

GradedModule

Date of creation

2013-03-22 11:45:05

Last modified on

2013-03-22 11:45:05

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

18

Author

CWoo (3771)

Entry type

Definition

Classification

msc 16W50

Classification

msc 74R99

Synonym

graded homomorphism

Synonym

homogeneous submodule

Related topic

GradedAlgebra

Defines

graded module homomorphism

Defines

homogeneous of degree

Defines

graded submodule

Defines

grading

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