Cryptography Reference
In-Depth Information
8.9 Galois Cohomology
In this section, we give the definition of the full Shafarevich-Tate group.
This requires reinterpreting and generalizing the descent calculations in terms
of Galois cohomology. Fortunately, we only need the first two cohomology
groups, and they can be defined in concrete terms.
Let
G
be a group and let
M
be an additive abelian group on which
G
acts.
This means that each
g
∈
G
gives a automorphism
g
:
M
→
M
.Moreover,
(
g
1
g
2
)(
m
)=
g
1
(
g
2
(
m
))
for all
m ∈ M
and all
g
1
,g
2
∈ G
.Weca lsuchan
M
a
G
-module
.One
possibility is that
g
is the identity map for all
g ∈ G
. In this case, we say that
the action of
G
is
trivial
.
If
G
is a topological group, and
M
has a topology, then we require that the
action of
G
on
M
be continuous. We also require all maps to be continuous.
In the cases below where the groups have topologies, this will always be the
case, so we will not discuss this point further.
A
homomorphism
φ
:
M
1
→
M
2
of
G
-modules
is a homomorphism of
abelian groups that is compatible with the action of
G
:
φ
(
gm
1
)=
gφ
(
m
1
)
for all
g ∈ G
and all
m
1
∈ M
1
. Note that
φ
(
m
1
)isanelementof
M
2
,so
gφ
(
m
1
) is the action of
g
on an element of
M
2
.An
exact sequence
0
→
M
1
→
M
2
→
M
3
→
0
is a short way of writing that the map from
M
1
to
M
2
is injective, the map from
M
2
to
M
3
is surjective, and the image of
M
1
→
M
2
is the kernel of
M
2
→
M
3
.
The most common situation is when
M
1
⊆ M
2
and
M
3
=
M
2
/M
1
.
More generally, a sequence of abelian groups and homomorphisms
···→A → B → C →···
is said to be
exact at
B
if the image of
A
C
. Such
a sequence is said to be
exact
if it is exact at each group in the sequence.
Define the
zeroth cohomology group
to be
→
B
is the kernel of
B
→
H
0
(
G, M
)=
M
G
=
{m ∈ M | gm
=
m
for all
g ∈ G}.
For example, if
G
acts trivially, then
H
0
(
G, M
)=
M
.
Define the
cocycles
Z
(
G, M
)=
{
maps
f
:
G → M | f
(
g
1
g
2
)=
f
(
g
1
)+
g
1
f
(
g
2
) for all
g
1
,g
2
∈ G}.
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