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152 7. Galois Cohomology and Isomorphism Classification of Elliptic Curves

δ
0
0
0
1
1
i ∗
i ∗
1 → H (G, A) → H (G, E) → H (G, E/A) → H (G, A) → H (G, E).
Further, if A is a normal subgroupof E, then
1
1
1
H (G, A) → H (G, E) → H (G, E/A)
is an exact sequence of pointed sets so that the entire six-term sequence is exact.
G
G
Proof. We have already shown that the image equals the kernel in A and E ,see
0
1
G
(2.6). For exactness at (E/A) = H (G, E/A) observe that δ(X) = 1in H (G, A)
is equivalent to the coset X having the form xA with s(x) = x for all s ∈ G by (3.8).
In other words, it is equivalent to the principal homogeneous G-set X being in the
0
0
image of H (G, E) → H (G, E/A).
1
1
To show exactness at H (G, A), we consider [a s ] ∈ H (G, A) whose image in
1
H (G, E) is trivial. This means for A ⊂ E that there exists e ∈ E with a s = e −1 s
( e)
s 0
for all s ∈ G. Then e = ea s andwehave[a s ] = δ(eA), where eA ∈ H (G, E/A) =
1
G
1
(E/A) . Clearly δ(eA) in H (G, A) has trivial image in H (G, E).
1
When A is a normal subgroup of E, we prove exactness at H (G, E). Let [a s ] ∈
1
1
H (G, E) have trivial image in H (G, E/A), that is, suppose there exists e ∈ E
with a s = e −1 s s s
( e)b s with b s ∈ A for all s ∈ G. Since ( e)A = A( e),wehave
c s ∈ A with
s
( e)b s = e
a s = e −1 s −1 c s e for all s ∈ G.
1
Then c s ∈ Z −1 (G, A) defining [c s ] ∈ H (G, A) which maps to [a s ] under i ∗ . This
proves the theorem.
i
(4.4) Remark. When A is commutative, it is possible to define H (G, A) for all
i ≥ 0 algebraically. The six-term sequence of (4.3) becomes seven term for A abelian
and normal in E. It has the form
0 i ∗ 0 0 δ 1 i ∗
1 → H (G, A) → H (G, E) → H (G, E/A) → H (G, A) →
δ
1
2
1
H (G, E) → H (G, E/A) → H (G, A).
If both A and E are commutative we have an exact triangle or long exact sequence
i
for all H of A, E,and E/A of the form
i ∗
∗
∗
H (G, A) −→ H (G, E)
H (G, E/A)
∗
where δ has degree +1.
(4.5) Remark. Let f : G → G be a group homomorphism. If E is a G-group

s s f (s )

with action x, then E is also a G -group with action x = x and f induces a
0
0
∗
natural morphism H (G, E) = E G → E G = H (G , E) denoted f and a natural

1
1
∗
morphism H (G, E) → H (G , E) also denoted f ([a s ]) = [a f (s ) ].