Morita equivalence

Morita equivalence

Let

R

be a ring. Write

ℳ

R

for the

category

of

right modules

over

R

. Two rings

R

and

S

are said to be

Morita equivalent

if

ℳ

R

and

ℳ

S

are

equivalent

as categories (

http://planetmath.org/EquivalenceOfCategories

). What this means is: we have two

functors

F

:

ℳ

R

→

ℳ

S

and

G

:

ℳ

S

→

ℳ

R

such that for any right

R

-module

M

and any right

S

-module

N

, we have

G

⁢

F

⁢

(

M

)

≅

R

M

and

F

⁢

G

⁢

(

N

)

≅

S

N

,

where

A

≅

R

B

means that there is an

R

-module

isomorphism

between

A

and

B

.

Example

. Any ring

R

with

1

is Morita equivalent to any

matrix ring

M

n

⁢

(

R

)

over it.

Proof.

Assume

n

>

1

. For convenience, we will also say a module to mean a right module.

Let

M

be an

R

-module. Set

F

⁢

(

M

)

=

{

(

m

1

,

…

,

m

n

)

∣

m

i

∈

M

}

. Then

F

⁢

(

M

)

becomes a module over

M

n

⁢

(

R

)

if we adopt the

standard matrix multiplication

m

⁢

A

, where

m

∈

F

⁢

(

M

)

and

A

∈

M

n

⁢

(

R

)

. If

f

:

M

1

→

M

2

is an

R

-

module homomorphism

. Set

F

⁢

(

f

)

:

F

⁢

(

M

1

)

→

F

⁢

(

M

2

)

by

F

⁢

(

f

)

⁢

(

m

1

,

…

,

m

n

)

=

(

f

⁢

(

m

1

)

,

…

,

f

⁢

(

m

n

)

)

∈

F

⁢

(

M

2

)

. Then

F

is a

covariant functor

by inspection.

Next, let

N

be an

M

n

⁢

(

R

)

-module. Write

e

⁢

(

r

)

as the

n

×

n

matrix whose cell

(

1

,

1

)

is

r

∈

R

and

0

everywhere else. For simplicity we write

e

:=

e

⁢

(

1

)

. Note that

e

is an

idempotent

in

M

n

⁢

(

R

)

:

e

=

e

⁢

e

, and

e

commutes with

e

⁢

(

r

)

for any

r

∈

R

:

e

⁢

e

⁢

(

r

)

=

e

⁢

(

r

)

⁢

e

.

Set

G

⁢

(

N

)

=

{

s

⁢

e

∣

s

∈

N

}

. For any

r

∈

R

, define

s

⁢

e

⋅

r

:=

s

⁢

e

⁢

e

⁢

(

r

)

=

s

⁢

e

⁢

(

r

)

⁢

e

. Since

s

⁢

e

⁢

(

r

)

∈

N

, this

multiplication

turns

G

⁢

(

N

)

into an

R

-module. If

g

:

N

1

→

N

2

is an

M

n

⁢

(

R

)

-module homomorphism, define

G

⁢

(

g

)

:

G

⁢

(

N

1

)

→

G

⁢

(

N

2

)

by

G

⁢

(

g

)

⁢

(

s

⁢

e

)

=

g

⁢

(

s

)

⁢

e

. If

N

1

⟶

g

N

2

⟶

h

N

3

are

M

n

⁢

(

R

)

-module homomorphisms, then

G

⁢

(

h

∘

g

)

⁢

(

s

⁢

e

)

=

(

h

∘

g

)

⁢

(

s

)

⁢

e

=

h

⁢

(

g

⁢

(

s

)

)

⁢

e

=

G

⁢

(

h

)

⁢

[

g

⁢

(

s

)

⁢

e

]

=

G

⁢

(

h

)

⁢

[

G

⁢

(

g

)

⁢

s

⁢

e

]

=

G

⁢

(

h

)

∘

G

⁢

(

g

)

⁢

(

s

⁢

e

)

so that

G

is a covariant functor.

If

M

is any

R

-module, then

G

⁢

F

⁢

(

M

)

=

{

(

m

1

,

…

,

m

n

)

⁢

e

∣

m

∈

M

}

=

{

(

m

1

,

0

,

…

,

0

)

T

∣

m

∈

M

}

≅

M

, where

m

T

stands for the

transpose

of the

row vector

m

∈

M

into a

column vector

.

On the other hand, if

N

is any

M

n

⁢

(

R

)

-module, then

F

⁢

G

⁢

(

N

)

=

{

(

s

1

⁢

e

,

…

,

s

n

⁢

e

)

∣

s

i

∈

N

}

. Before proving that

F

⁢

G

⁢

(

N

)

≅

N

, let’s do some preliminary work.

Denote

e

i

⁢

i

by the

n

×

n

matrix whose cell

(

i

,

i

)

is 1 and

0

everywhere else. Then each

e

i

⁢

i

is idempotent,

e

i

⁢

i

⁢

e

j

⁢

j

=

0

for

i

≠

j

, and

e

11

+

⋯

+

e

n

⁢

n

=

1

. From this, we see that

N

=

N

1

⊕

⋯

⊕

N

n

, where

N

i

=

N

⁢

e

i

⁢

i

, and

N

i

≅

N

j

as

M

n

⁢

(

R

)

-modules. Since

N

1

=

N

⁢

e

has an

R

-module

structure

as we had shown earlier,

N

i

are all

R

-modules. Let

π

i

:

N

→

N

i

be the

projection map

,

ψ

i

:

N

i

→

N

be the

embedding

of

N

i

into

N

, and

ϕ

i

⁢

j

:

N

i

→

N

j

be the isomorphism from

N

i

to

N

j

given by

ϕ

i

⁢

j

⁢

(

s

⁢

e

i

⁢

i

)

=

s

⁢

e

j

⁢

j

. All these are

M

n

⁢

(

R

)

-module homomorphisms since

e

i

⁢

i

⁢

A

=

A

⁢

e

i

⁢

i

.

Now, take any

s

∈

N

, then

s

↦

(

π

1

⁢

(

s

)

,

…

,

π

n

⁢

(

s

)

)

↦

(

ϕ

11

⁢

π

1

⁢

(

s

)

,

…

,

ϕ

n

⁢

1

⁢

π

n

⁢

(

s

)

)

∈

F

⁢

G

⁢

(

N

)

is a

homomorphism

α

:

N

→

F

⁢

G

⁢

(

N

)

.

Conversely

,

(

s

1

⁢

e

,

…

,

s

n

⁢

e

)

↦

(

ϕ

11

⁢

(

s

1

⁢

e

)

,

…

,

ϕ

1

⁢

n

⁢

(

s

n

⁢

e

)

)

↦

ψ

1

⁢

(

ϕ

11

⁢

(

s

1

⁢

e

)

)

+

⋯

+

ψ

n

⁢

(

ϕ

1

⁢

n

⁢

(

s

n

⁢

e

)

)

∈

N

is also a homomorphism

β

:

F

⁢

G

⁢

(

N

)

→

N

. By inspection,

α

and

β

are

inverses

of each other, and hence

F

⁢

G

⁢

(

N

)

≅

N

.
∎

Remark

. A property

P

in the class of all rings is said to be

Morita invariant

if, whenever

R

has property

P

and

S

is Morita equivalent to

R

, then

S

has property

P

as well. By the example above, it is clear that

commutativity

is not a Morita invariant property.

Title

Morita equivalence

Canonical name

MoritaEquivalence

Date of creation

2013-03-22 16:38:49

Last modified on

2013-03-22 16:38:49

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Definition

Classification

msc 16D90

Defines

Morita equivalent

Defines

Morita invariance

Defines

Morita invariant

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