Jacobson radical of a module category and its power

Jacobson radical of a module category and its power

Assume that

k

is a field and

A

is a

k

-

algebra

. The

category

of (left)

A

-modules will be denoted by

Mod

⁢

(

A

)

and

Hom

A

⁢

(

X

,

Y

)

will denote the set of all

A

-

homomorphisms

between

A

-modules

X

and

Y

. Of course

Hom

A

⁢

(

X

,

Y

)

is an

A

-module itself and

End

A

⁢

(

X

)

=

Hom

A

⁢

(

X

,

X

)

is a

k

-algebra (even

A

-algebra) with

composition

as a

multiplication

.

Let

X

and

Y

be

A

-modules. Define

rad

A

⁢

(

X

,

Y

)

=

{

f

∈

Hom

A

⁢

(

X

,

Y

)

|

∀

g

∈

Hom

A

⁢

(

Y

,

X

)

1

X

-

g

⁢

f

⁢

is invertible in

⁢

End

A

⁢

(

X

)

}

.

Definition.

The

Jacobson radical

of a category

Mod

⁢

(

A

)

is defined as a class

rad

⁢

Mod

⁢

(

A

)

=

⋃

X

,

Y

∈

Mod

⁢

(

A

)

rad

A

⁢

(

X

,

Y

)

.

□

Properties.

1

)

The Jacobson radical is an ideal in

Mod

⁢

(

A

)

, i.e. for any

X

,

Y

,

Z

∈

Mod

⁢

(

A

)

, for any

f

∈

rad

A

⁢

(

X

,

Y

)

, any

h

∈

Hom

A

⁢

(

Y

,

Z

)

and any

g

∈

Hom

A

⁢

(

Z

,

X

)

we have

h

⁢

f

∈

rad

A

⁢

(

X

,

Z

)

and

f

⁢

g

∈

rad

A

⁢

(

Z

,

X

)

. Additionaly

rad

A

⁢

(

X

,

Y

)

is an

A

-

submodule

of

Hom

A

⁢

(

X

,

Y

)

.

2

)

For any

A

-module

X

we have

rad

A

⁢

(

X

,

X

)

=

rad

⁢

(

End

A

⁢

(

X

)

)

, where on the right side we have the classical Jacobson radical.

3

)

If

X

,

Y

are both

indecomposable

A

-modules such that both

End

A

⁢

(

X

)

and

End

A

⁢

(

Y

)

are local algebras (in particular, if

X

and

Y

are

finite dimensional

), then

rad

A

⁢

(

X

,

Y

)

=

{

f

∈

Hom

A

⁢

(

X

,

Y

)

|

f

⁢

is not an isomorphism

}

.

In particular, if

X

and

Y

are not

isomorphic

, then

rad

A

⁢

(

X

,

Y

)

=

Hom

A

⁢

(

X

,

Y

)

.

□

Let

n

∈

ℕ

and let

f

∈

Hom

A

⁢

(

X

,

Y

)

. Assume there is a

sequence

of

A

-modules

X

=

X

0

,

X

1

,

…

,

X

n

-

1

,

X

n

=

Y

and for any

0

≤

i

≤

n

-

1

we have an

A

-homomorphism

f

i

∈

rad

A

⁢

(

X

i

,

X

i

+

1

)

such that

f

=

f

n

-

1

⁢

f

n

-

2

⁢

⋯

⁢

f

1

⁢

f

0

. Then we will say that

f

is

n

-factorizable through Jacobson radical

.

Definition.

The

n

-th power of a Jacobson radical

of a category

Mod

⁢

(

A

)

is defined as a class

rad

n

⁢

Mod

⁢

(

A

)

=

⋃

X

,

Y

∈

Mod

⁢

(

A

)

rad

A

n

⁢

(

X

,

Y

)

,

where

rad

A

n

⁢

(

X

,

Y

)

is an

A

-submodule of

Hom

A

⁢

(

X

,

Y

)

generated by all homomorphisms

n

-factorizable through Jacobson radical. Additionaly define

rad

A

∞

⁢

(

X

,

Y

)

=

⋂

n

=

1

∞

rad

A

n

⁢

(

X

,

Y

)

⁢

□

Properties.

0

)

Obviously

rad

A

⁢

(

X

,

Y

)

=

rad

A

1

⁢

(

X

,

Y

)

and for any

n

∈

ℕ

we have

rad

A

n

⁢

(

X

,

Y

)

⊇

rad

A

∞

⁢

(

X

,

Y

)

.

1

)

Of course each

rad

A

n

⁢

(

X

,

Y

)

is an

A

-submodule of

Hom

A

⁢

(

X

,

Y

)

and we have following sequence of

inclusions

:

Hom

A

⁢

(

X

,

Y

)

⊇

rad

A

1

⁢

(

X

,

Y

)

⊇

rad

A

2

⁢

(

X

,

Y

)

⊇

rad

A

3

⁢

(

X

,

Y

)

⊇

⋯

2

)

If both

X

and

Y

are finite dimensional, then there exists

n

∈

ℕ

such that

rad

A

∞

⁢

(

X

,

Y

)

=

rad

A

n

⁢

(

X

,

Y

)

.

Title

Jacobson radical of a module category and its power

Canonical name

JacobsonRadicalOfAModuleCategoryAndItsPower

Date of creation

2013-03-22 19:02:23

Last modified on

2013-03-22 19:02:23

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

6

Author

joking (16130)

Entry type

Definition

Classification

msc 16N20

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