Suppose is a subgroup of a finite group and
is a -module. For each , there is a natural map

called restriction. Elements of can be viewed as classes
of -cocycles, which are certain maps
, and the restriction maps restricts these cocycles to
.
If is a normal subgroup of , there is also an inflation map

given by taking a cocycle
and precomposing with the quotient map to obtain a cocycle for .

*Proof*.
Our proof follows [

Ser79, pg. 117] closely.

We see that
by looking at cochains. It remains
to prove that
is injective and that the image of
is the kernel of .

*That
is injective:* Suppose
is a
cocycle whose image in is equivalent to 0 modulo
coboundaries. Then there is an such that
, where we identify with the map that is
constant on the costs of . But depends only on the costs of
modulo , so
for all
, i.e.,
(as we see by adding to both
sides and multiplying by
).Thus , so is
equivalent to 0 in
.

*The image of
contains the kernel of :*
Suppose is a cocycle whose
restriction to is a coboundary, i.e., there is such
that
for all
.
Subtracting the coboundary
for
from , we may assume
for all
.
Examing the equation
with shows that is constant on the cosets of .
Again using this formula, but with
and , we see
that

so the image of is contained in . Thus defines a cocycle
, i.e., is in the image of
.

This proposition will be useful when proving
the weak Mordell-Weil theorem.

*Example 11.3.2*
The sequence of Proposition

11.3.1 need not be
surjective on the right. For example, suppose

,
and let

act trivially on the cyclic group

.
Using the

interpretation of

, we see
that

, but

has order

.

*Remark 11.3.3*
On generalization of Proposition

11.3.1 is to
a more complicated exact sequence involving the ``transgression map''
tr:

Another generalization of Proposition

11.3.1
is that if

for

, then
there is an exact sequence

*Remark 11.3.4*
If

is a not-necessarily-normal subgroup of

, there are also
maps

for each

. For

this is the trace map

, but the definition for

is more involved.
One has

.
Taking

we see
that for each

the group

is annihilated by

.

William Stein
2012-09-24