Tate's duality maps through a normal subgroup of finite index, by dimension shifting

Let G be a compact topological group, V a normal subgroup of finite index, and N a discrete G-module with N ≃+ ZMod n and H²(G, N) ≃+ ZMod n, on which V acts trivially. For a finite discrete G-module M write M' = InternalHom G M N for its dual and αᵢ : Hⁱ(G, M) → Hom(H²⁻ⁱ(G, M'), H²(G, N)), i = 0, 1, 2, for Tate's duality maps (TauCeti.ContCohomology.dualityMap0, dualityMap1, dualityMap2).

This file proves that Tate's duality on the coinduced modules gives Tate's duality on every finite module killed by n on which V acts trivially (TauCeti.ContCohomology.dualityMap0_bijective_of_bijective_discreteCoind, dualityMap1_bijective_of_injective_discreteCoind, dualityMap2_bijective_of_bijective_discreteCoind): if α₀, α₁ are bijective and α₂ is injective on Coind_V^G A for every such A, then α₀ and α₂ are bijective on every such A; and if α₀ is bijective and α₁ injective on Coind_V^G A for every such A, then so is α₁ on every such A once H¹(G, A') is finite. Over a local field K, with V = G_L for a finite Galois extension L/K containing the n-th roots of unity and N = μₙ, this is the step of local Tate duality that passes from the coinduced modules, whose duality is that of G_L by Shapiro's lemma, to all finite Galois modules killed by n that become trivial over L.

The class 𝒞 of finite discrete G-modules killed by n on which V acts trivially is closed under the two operations the argument needs: the cokernel Coind_V^G A ⧸ A of the unit of coinduction (TauCeti.ContCohomology.CoindQuotient.smul_eq_self_of_forall_smul_eq_self, using normality of V), and the dual A', since V acts trivially on N. The argument runs in two passes over 𝒞, along the short exact sequence 0 → A → Coind_V^G A → C → 0 of the unit (TauCeti.ContCohomology.coindShortExact), whose dual sequence exists for all coefficients (TauCeti.ContCohomology.precomp_coindShortExact_inclDistribMulActionHom_surjective).

• Injectivity, degree by degree. α₀ is injective on A because it is on Coind_V^G A and H⁰ is left exact (TauCeti.ContCohomology.dualityMap0_injective_of_injective). Injectivity of αᵢ on the cokernel C, which lies in 𝒞, then gives injectivity of αᵢ₊₁ on A by the four lemma along the ladder of the sequence (TauCeti.ContCohomology.DiscreteShortExact.dualityMap1_injective_left, DiscreteShortExact.dualityMap2_injective_left); the Baer hypothesis on H²(G, N) holds since ℤ/nℤ is self-injective.
• Bijectivity, by counting. On A ∈ 𝒞 with dual A' ∈ 𝒞, injectivity of αᵢ on A and of α₂₋ᵢ on A' forces |Hⁱ(G, A)| = |H²⁻ⁱ(G, A')|, so that αᵢ is bijective (TauCeti.ContCohomology.dualityMap0_bijective_of_injective_of_addEquiv_zmod and its companions). Finiteness of H¹(G, A') is the one input of the count not supplied by the argument itself: H⁰ is a subgroup of a finite module, and H²(G, A') embeds by α₂ into the finite group Hom(H⁰(G, A''), H²(G, N)).

Main results

• TauCeti.ContCohomology.dualityMap0_bijective_of_bijective_discreteCoind, dualityMap1_bijective_of_injective_discreteCoind and dualityMap2_bijective_of_bijective_discreteCoind: Tate's duality on the finite modules killed by n on which V acts trivially, from Tate's duality on their coinduced modules (in degree 1, only α₀ bijective and α₁ injective there, together with the finiteness of H¹ of the dual).

References

• J.-P. Serre, Galois Cohomology, Ch. II, §5.2, proof of Theorem 2.
• J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields, 2nd ed., (7.2.6).

theorem TauCeti.ContCohomology.dualityMap0_bijective_of_bijective_discreteCoind {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] {V : Subgroup G} [V.Normal] [V.FiniteIndex] {N : Type u} [AddCommGroup N] [TopologicalSpace N] [DiscreteTopology N] [DistribMulAction G N] [ContinuousSMul G N] {n : ℕ} [NeZero n] (e : N ≃+ ZMod n) (e₂ : H2 G N ≃+ ZMod n) (hN : ∀ v ∈ V, ∀ (y : N), v • y = y) (hcoind : ∀ (A : Type u) [inst : AddCommGroup A] [inst_1 : TopologicalSpace A] [DiscreteTopology A] [inst_3 : DistribMulAction G A] [ContinuousSMul G A] [inst_5 : Finite A], (∀ (a : A), n • a = 0) → (∀ v ∈ V, ∀ (a : A), v • a = a) → Function.Bijective ⇑(dualityMap0 G (DiscreteCoind G V A) N) ∧ Function.Bijective ⇑(dualityMap1 G (DiscreteCoind G V A) N) ∧ Function.Injective ⇑(dualityMap2 G (DiscreteCoind G V A) N)) (A : Type u) [AddCommGroup A] [TopologicalSpace A] [DiscreteTopology A] [DistribMulAction G A] [ContinuousSMul G A] [Finite A] (hA : ∀ (a : A), n • a = 0) (hAV : ∀ v ∈ V, ∀ (a : A), v • a = a) : Function.Bijective ⇑(dualityMap0 G A N)

Tate's duality map α₀ through a normal subgroup of finite index. Let G be a compact group, V a normal subgroup of finite index, and N a discrete G-module on which V acts trivially, with N ≃+ ZMod n and H²(G, N) ≃+ ZMod n. Suppose that for every finite discrete G-module A killed by n on which V acts trivially, Tate's duality maps α₀, α₁ are bijective and α₂ is injective on the coinduced module Coind_V^G A. Then α₀ : H⁰(G, A) → Hom(H²(G, A'), H²(G, N)) is bijective on every such A.

theorem TauCeti.ContCohomology.dualityMap1_bijective_of_injective_discreteCoind {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] {V : Subgroup G} [V.Normal] [V.FiniteIndex] {N : Type u} [AddCommGroup N] [TopologicalSpace N] [DiscreteTopology N] [DistribMulAction G N] [ContinuousSMul G N] {n : ℕ} [NeZero n] (e : N ≃+ ZMod n) (e₂ : H2 G N ≃+ ZMod n) (hN : ∀ v ∈ V, ∀ (y : N), v • y = y) (hcoind₁ : ∀ (A : Type u) [inst : AddCommGroup A] [inst_1 : TopologicalSpace A] [DiscreteTopology A] [inst_3 : DistribMulAction G A] [ContinuousSMul G A] [inst_5 : Finite A], (∀ (a : A), n • a = 0) → (∀ v ∈ V, ∀ (a : A), v • a = a) → Function.Bijective ⇑(dualityMap0 G (DiscreteCoind G V A) N) ∧ Function.Injective ⇑(dualityMap1 G (DiscreteCoind G V A) N)) (A : Type u) [AddCommGroup A] [TopologicalSpace A] [DiscreteTopology A] [DistribMulAction G A] [ContinuousSMul G A] [Finite A] (hA : ∀ (a : A), n • a = 0) (hAV : ∀ v ∈ V, ∀ (a : A), v • a = a) [Finite (H1 G (InternalHom G A N))] : Function.Bijective ⇑(dualityMap1 G A N)

Tate's duality map α₁ through a normal subgroup of finite index. With G, V, N as in TauCeti.ContCohomology.dualityMap0_bijective_of_bijective_discreteCoind, suppose that for every finite discrete G-module A killed by n on which V acts trivially, α₀ is bijective and α₁ is injective on the coinduced module Coind_V^G A. If moreover H¹(G, A') is finite, then α₁ : H¹(G, A) → Hom(H¹(G, A'), H²(G, N)) is bijective on every such A.

theorem TauCeti.ContCohomology.dualityMap2_bijective_of_bijective_discreteCoind {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] {V : Subgroup G} [V.Normal] [V.FiniteIndex] {N : Type u} [AddCommGroup N] [TopologicalSpace N] [DiscreteTopology N] [DistribMulAction G N] [ContinuousSMul G N] {n : ℕ} [NeZero n] (e : N ≃+ ZMod n) (e₂ : H2 G N ≃+ ZMod n) (hN : ∀ v ∈ V, ∀ (y : N), v • y = y) (hcoind : ∀ (A : Type u) [inst : AddCommGroup A] [inst_1 : TopologicalSpace A] [DiscreteTopology A] [inst_3 : DistribMulAction G A] [ContinuousSMul G A] [inst_5 : Finite A], (∀ (a : A), n • a = 0) → (∀ v ∈ V, ∀ (a : A), v • a = a) → Function.Bijective ⇑(dualityMap0 G (DiscreteCoind G V A) N) ∧ Function.Bijective ⇑(dualityMap1 G (DiscreteCoind G V A) N) ∧ Function.Injective ⇑(dualityMap2 G (DiscreteCoind G V A) N)) (A : Type u) [AddCommGroup A] [TopologicalSpace A] [DiscreteTopology A] [DistribMulAction G A] [ContinuousSMul G A] [Finite A] (hA : ∀ (a : A), n • a = 0) (hAV : ∀ v ∈ V, ∀ (a : A), v • a = a) : Function.Bijective ⇑(dualityMap2 G A N)

Tate's duality map α₂ through a normal subgroup of finite index: under the hypotheses of TauCeti.ContCohomology.dualityMap0_bijective_of_bijective_discreteCoind, α₂ : H²(G, A) → Hom(H⁰(G, A'), H²(G, N)) is bijective.