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

14 Complete resolution of G
Recall our earlier definition of P•, giving a free Z[G]-resolution of Z. Dualising
this by defining P ∗
i = HomZ(Pi, Z) over Z, not Z[G], we obtain a second
exact sequence
· · ·
P2
P1
P0
0
ε
Z
0
ε
Z ∗
P ∗ 0
P ∗ 1
P ∗ 2
· · ·
We extend this to negative indices by setting P−n = P ∗ n−1 . Thus, we obtain
a long exact sequence
called a complete resolution.
. . . → P2 → P1 → P0 → P−1 → P−2 → . . .
Proposition 42. ˆ H q (G, A) is the q-th cohomology group of the complex
and we have an exact sequence
· · · ← Hom(P1, A) ← Hom(P0, A) ← Hom(P−1, A) ← Hom(P−2, A) ← · · · .
Proof. In the case q ≥ 1, this follows immediately from the definition.
Suppose now that q ≤ −2. By definition, Hn(G, A) is the nth homology
of the complex P•⊗ Z[G]A, i.e., the standard complex, although any projective
resolution of Z will work. But for a finitely generated free Z[G]-module B,
B × A ∼ −→ Hom(B ∗ , A), c × a ↦→ (f ↦→ f(c).a),
where B ∗ = Hom(B, Z), is an isomorphism of Z[G]-modules. So we obtain
an isomorphism τ
τ : B ⊗Z[G] A = (B ⊗ A)/IG.(B ⊗ A) ∼
−−→ (B ⊗ A)G
N ∗
→ Hom(B ∗ , A) G = HomG(B ∗ , A),
which can be verified using the identities b⊗s.a = s −1 .b⊗a, s.b⊗s.a = b⊗a,
and (s − 1).(b ⊗ a) = s.b ⊗ s.a − b ⊗ a. Moreover, we note that N ∗ is an
isomorphism since B ⊗Z A is induced, which is the case because B is free
and the tensor product is formed over Z. Thus
HomG(P−n, A) = HomG(Hom(Pn−1, Z), A) ∼ = Pn−1 ⊗ Z[G] A
and the result follows since we are computing cohomology of the same complex.
For the cases q = 0 and q = 1, see Cassels–Fröhlich, p. 103.
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