A Short Course on Galois Cohomology - William Stein - University of ...

But in fact it induces multiplication by −1 on A:
s = t −1 st
s
G
G
A
A
a
t−1a = −a
Thus it induces the map H q (G, A) −1
−→ H q (G, A), and we conclude that, on
the k-vector space H q (G, A), we have −1 = 1 and so H q (G, A) = 0 for all
q ≥ 0.
Remark 26. In the proof we can replace G by any subgroup of GLn(k)
that contains a non-identity scalar t.
Example 27. Let E/k be an ellipctic curve and suppose that ¯ρE,p : Gk ↠
Aut(E[p]) is surjective. Then
for all q ≥ 0.
H q (Gal(k(E[p])/k), E[p]) = 0
9 The restriction-inflation sequence
Proposition 28. Let H ⊳ G be a normal subgroup and A ∈ ModG. Then
we have an exact sequence
0 → H 1 (G/H, A H ) inf
−→ H 1 (G, A) res
−→ H 1 (H, A)
Proof. We check this directly on 1-cocycles, following Atiyah–Wall. We begin
with exactness at H1 (G/H, AH ), that is, the injectivity of inf. Consider
H1 (G/H, AH ) inf
−→ H1 (G, A), [f] ↦→ [ ¯ f] where
G
G/H f
¯f
14
A
AH