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

6 Inflation and restriction
Suppose that H is a subgroup of G and let A ∈ ModG. A morphism of
pairs (G, A) → (H, A) by H → G and id: A → A induces a restriction
homomorphism
resH : H q (G, A) → H q (H, A).
For example, in terms of 1-cocyles, we see that the homomorphism
resH : H 1 (G, A) → H 1 (H, A), [f] ↦→ [f|H] is given by the usual restriction
map.
Next suppose that H ⊳ G. We obtain a morphism of pairs (G/H, A H ) →
(G, A). This induces an inflation homomorphism
inf : H q (G/H, A H ) → H q (G, A).
We shall later consider the following theorem, whose full proof uses spectral
sequences:
Theorem 23. The sequence
0 → H 1 (G/H, A H ) inf
−→ H 1 (G, A) res
−→ H 1 (H, A) → H 2 (G/H, A H ) → . . .
is exact.
7 Inner automorphisms
Suppose t ∈ G. The conjugation s ↦→ tst −1 is an inner automorphism of
G. This induces a morphism of pairs (G, A) → (G, A): G ← G, tst −1 ← s,
A → A, a ↦→ t −1 a.
We obtain a homomorphism H q (G, A) → H q (G, A), that is, a natural
action of G on H q (G, A).
Proposition 24. This action is trivial, that is, the map t ∈ G induces the
identity map on H q (G, A).
12