Applying the basic exact sequence.

Let K be an “LF” as usual, with residue field k of characteristic p, K̄/K an algebraic closure and G_{K} = Gal (K̄/K). Let M be a finite abelian group with a continuous G_{K}- action. Let’s examine the basic exact sequence:

(∗) 0 → H^{1}(G_{k}, M^{I_{k}}) → H^{1}(G_{K}, M) → H^{1}(I_{K}, M)^{G_{k}} → 0,

and let us “evaluate” this sequence in even more down-to-earth terms in special instances:

1. If M is of order prime to p, and the G_{K} action on M is tamely ramified. Then the wild ramification group P_{K} ⊂ I_{K} acts trivially on M; therefore G_{K} acts through the tame quotient T_{K} and I_{K} acts through the quotient group Δ_{K} = I_{K}/P_{K}, which fits in the exact sequence:

0 → Δ_{K} → T_{K} → G_{k} → 0.

Under our hypotheses, the natural homomorphism induces an isomorphism of cohomology groups H^{1}(Δ_{K}, M) ≅ H^{1}(I_{K}, M) (for we have an exact sequence 0 → H^{1}(Δ_{K}, M^{P_{K}}) → H^{1}(I_{K}, M) → H^{1}(P_{K}, M) and the last of these groups vanishes).

Therefore our basic sequence (∗) becomes:

(∗∗) 0 → H^{1}(G_{k}, M^{Δ_{k}}) → H^{1}(G_{K}, M) → H^{1}(Δ_{K}, M)^{G_{k}} → 0.

Since both G_{k} and Δ are canonically Ẑ = ∏_{ℓ} Z_{ℓ} and ∏_{ℓ≠p} Z_{ℓ}(1) respectively, we may make use of the canonical isomorphism evaluating H^{1} of such a group (call it C) acting continuously on a finite abelian goup N,

C ⊗ H^{1}(C, N) ≅ N_{C}

to evaluate the flanking terms in this exact sequence as:

0 → (M^{Δ_{K}})_{G_{k}} → H^{1}(G_{K}, M) → (M_{Δ_{K}} (−1))^{G_{k}} → 0.

Here we should explain the “(−1)” carefully.

Let’s compare this evaluation with Tate Local Duality. So, let us consider the Cartier Dual of M,

M^{∗} = Hom (M, μ(K̄)) = M^{∧} ⊗ Ta(μ(K̄)).

Here M^{∧} is the pontrjagin dual of M; i.e., M^{∧} = Hom (M, Q/Z), and Ta(μ(K̄) is the Tate module of μ(K̄); i.e., the module such that there is a canonical isomorphism

Ta(μ(K̄)) ⊗ Q/Z ≅ μ(K̄).

So we have the exact sequences:

0 → (M^{Δ_{K}})_{G_{k}} → H^{1}(G_{K}, M) → (M_{Δ_{K}} (−1))^{G_{k}} → 0.

0 → ((M^{∗})^{Δ_{K}})_{G_{k}} → H^{1}(G_{K}, M^{∗}) → ((M^{∗})_{Δ_{K}} (−1))^{G_{k}} → 0.

To compare these more easily, let us pontrjagin-dualize the first exact sequence,

0 → [(M_{Δ_{K}} (−1))^{G_{k}}]^{∧} → [H^{1}(G_{K}, M)]^{∧} → [(M^{Δ_{K}})_{G_{k}}]^{∧} → 0.

Exercise. Unravelling the definitions, and using the standard properties of pontrjagin duality, find the tautological isomorphisms

(1) α : ((M^{∗})^{Δ_{K}})_{G_{k}}  ≅ [(M_{Δ_{K}} (−1))^{G_{k}}]^{∧},

and

(2) β : ((M^{∗})_{Δ_{K}} (−1))^{G_{k}}  ≅ [(M^{Δ_{K}})_{G_{k}}]^{∧}.

Proposition. The Tate Duality Isomorphism H^{1}(G_{K}, M^{∗}) ≅ [H^{1}(G_{K}, M)]^{∧} restricts to an isomorphism ((M^{∗})^{Δ_{K}})_{G_{k}} ≅ [(M_{Δ_{K}} (−1))^{G_{k}}]^{∧}, and induces an isomorphism on the quotient groups, ((M^{∗})_{Δ_{K}} (−1))^{G_{k}} ≅ [(M^{Δ_{K}})_{G_{k}}]^{∧}.

—-(Give proof)—-

1. If M is unramified (but no condition on its order) then the basic exact sequence reads:

0 → H^{1}(G_{k}, M) → H^{1}(G_{K}, M) → Hom_{G_{k}} (I_{K})^{ab}, M) → 0.

The finite and singular parts of 1-dimensional cohomology. If we are in either caser 1. or 2. above, let us give alternate notation for the two flanking groups in the basic exact sequence (*). Put:

H_{f}^{1}(G_{K}, M) := H^{1}(G_{k}, M),

and

H_{s}^{1}(G_{K}, M) := H^{1}(I_{K}, M)^{G_{k}},

and I will refer to these groups, respectively, as the finite part and the singular part of H_{f}^{1}(G_{K}, M), so that we can rewrite our basic exact sequence as:

0 → H_{f}^{1}(G_{K}, M) → H^{1}(G_{K}, M) → H_{s}^{1}(G_{K}, M) → 0,

and we have that Tate Duality, puts the finite part of H^{1}(G_{K}, M) in pontrijagin duality with the singular part of H^{1}(G_{K}, M^{∗}), and, of course, vice versa.