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

is an isomorphism. Reducing modulo v we have an exact sequence
0 → A 0 k → Ak → ΦA,v → 0, (20.1)
where A0 k is the connected component that contains the identity and ΦA,v
is a finite flat group scheme over k, called the component group of A at v.
Proposition 20.9. For every integer q, we have
ˆH q (k, Ak) = ˆ H q (k, ΦA,v).
Proof. Take Galois cohomology associated to the exact sequence (20.1), and
use Corollary 20.7.
21 Duality
WARNING: For the rest of this book, we’re going to let ¯ k denote a separable
closure of k, since it’s much easier notation to work with (I’ll go back
and change the notation above later).
Let k be a field and ¯ k a choice of separable closure of k.
21.1 Duality over a Local Field
Let M be any Gk = Gal( ¯ k/k)-module and set
ˆM = Hom(M, ¯ k ∗ ),
which we give the structure of (left) Gk-module by
(g.ϕ)(a) = g(ϕ(g −1 a)).
To see that this gives ˆ M a Gk-module structure, note that if g, h ∈ GK,
then
((gh).ϕ)(a) = (gh)(ϕ((gh) −1 a)) = (gh)(ϕ(h −1 g −1 a))
and
(g.(h.ϕ))(a) = g((h.ϕ)(g −1 a)) = g(h(ϕ(h −1 g −1 a))).
Theorem 21.1 (Tate Local Duality). Let k be a local field and M a finite
Gk-module of order coprime to the characteristic of k. Then for r = 0, 1, 2,
the cup product pairing
H r (k, M) × H 2−r (k, ˆ M) → H 2 (k, ¯ k ∗ ) ∼ = Q/Z
is nondegenerate. Also H q (k, M) = 0 for q ≥ 3.
The proof of this theorem is beyond the scope of this course, since it
requires developing too much general machinery. See [Ser97] for complete
details.
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