group cohomology

group cohomology

Let

G

be a group and let

M

be a (left)

G

-module. The

0

t

⁢

h

cohomology group

of the

G

-module

M

is

H

0

⁢

(

G

,

M

)

=

{

m

∈

M

:

∀

σ

∈

G

,

σ

⁢

m

=

m

}

which is the set of elements of

M

which are

G

-invariant, also
denoted by

M

G

.

A map

ϕ

:

G

→

M

is said to be a

crossed
homomorphism

(or

1-cocycle

) if

ϕ

⁢

(

α

⁢

β

)

=

ϕ

⁢

(

α

)

+

α

⁢

ϕ

⁢

(

β

)

for all

α

,

β

∈

G

. If we fix

m

∈

M

, the map

ρ

:

G

→

M

defined by

ρ

⁢

(

α

)

=

α

⁢

m

-

m

is clearly a crossed homomorphism, said to be

principal

(or

1-coboundary

). We define the following groups:

Z

1

⁢

(

G

,

M

)

=

{

ϕ

:

G

→

M

:

ϕ

is a 1-cocycle

}

B

1

⁢

(

G

,

M

)

=

{

ρ

:

G

→

M

:

ρ

is a
1-coboundary

}

Finally, the

1

s

⁢

t

cohomology group

of the

G

-module

M

is defined to be the

quotient group

:

H

1

⁢

(

G

,

M

)

=

Z

1

⁢

(

G

,

M

)

/

B

1

⁢

(

G

,

M

)

The following

proposition

is very useful when trying to compute
cohomology groups:

Proposition 1

.

Let

G

be a group and let

A

,

B

,

C

be

G

-modules related by an

exact sequence

:

0

→

A

→

B

→

C

→

0

Then there is a

long exact sequence in cohomology

:

0

→

H

0

⁢

(

G

,

A

)

→

H

0

⁢

(

G

,

B

)

→

H

0

⁢

(

G

,

C

)

→

H

1

⁢

(

G

,

A

)

→

H

1

⁢

(

G

,

B

)

→

H

1

⁢

(

G

,

C

)

→

…

In general, the cohomology groups

H

n

⁢

(

G

,

M

)

can be defined as
follows:

Definition 1

.

Define

C

0

⁢

(

G

,

M

)

=

M

and for

n

≥

1

define the

additive group

:

C

n

⁢

(

G

,

M

)

=

{

ϕ

:

G

n

→

M

}

The elements of

C

n

⁢

(

G

,

M

)

are called

n

-cochains. Also, for

n

≥

0

define the

n

t

⁢

h

coboundary

homomorphism

d

n

:

C

n

⁢

(

G

,

M

)

→

C

n

+

1

⁢

(

G

,

M

)

:

d

n

⁢

(

ϕ

)

⁢

(

g

1

,

…

,

g

n

+

1

)

=

g

1

⋅

ϕ

⁢

(

g

2

,

…

,

g

n

+

1

)

+

∑

i

=

1

n

(

-

1

)

i

⁢

ϕ

⁢

(

g

1

,

…

,

g

i

-

1

,

g

i

⁢

g

i

+

1

,

g

i

+

2

,

…

,

g

n

+

1

)

+

(

-

1

)

n

+

1

⁢

ϕ

⁢

(

g

1

,

…

,

g

n

)

Let

Z

n

⁢

(

G

,

M

)

=

ker

⁡

d

n

for

n

≥

0

, the set of

n

-cocyles. Also, let

B

0

⁢

(

G

,

M

)

=

1

and for

n

≥

1

let

B

n

⁢

(

G

,

M

)

=

image

⁡

d

n

-

1

, the set of

n

-coboundaries.

Finally we define the

n

t

⁢

h

-cohomology group of

G

with
coefficients in

M

to be

H

n

⁢

(

G

,

M

)

=

Z

n

⁢

(

G

,

M

)

/

B

n

⁢

(

G

,

M

)

References

1

J.P. Serre,

Galois Cohomology

,
Springer-Verlag, New York.

2

James Milne,

Elliptic Curves

.

3

Joseph H. Silverman,

The

Arithmetic of Elliptic Curves

. Springer-Verlag, New York, 1986.

Title

group cohomology

Canonical

name

GroupCohomology

Date of creation

2013-03-22 13:50:07

Last modified on

2013-03-22 13:50:07

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

11

Author

alozano (2414)

Entry type

Definition

Classification

msc 20J06

Synonym

cohomology

Related topic

SelmerGroup

Related topic

CohomologyGroupTheorem

Related topic

ProofOfCohomologyGroupTheorem

Related topic

OmegaSpectrum

Related topic

NaturalEquivalenceOfC_GAndC_MCategories

Defines

group cohomology

Defines

coboundary

Defines

cocycle

Defines

crossed homomorphism

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