Tate's duality maps of a Demushkin group with twisted coefficients

Let G be an infinite Demushkin group with canonical character χ = demushkinCharacter hG, and let I(χ)/pⁱ = TauCeti.ZModTwist χ i be the twisted coefficients ℤ/pⁱ on which G acts through χ, so that H²(G, I(χ)/pⁱ) is cyclic of order pⁱ (TauCeti.IsDemushkin.isAddCyclic_H2_zModTwist_demushkinCharacter). For a finite discrete G-module M, Tate's duality maps with these coefficients are

αⱼ : Hʲ(G, M) → Hom(H²⁻ʲ(G, Hom(M, I(χ)/pⁱ)), H²(G, I(χ)/pⁱ)), j = 0, 1, 2,

the cup products with the evaluation pairing (TauCeti.ContCohomology.dualityMap0, dualityMap1, dualityMap2). This file proves Tate's duality at level pⁱ: the three maps are bijective on every finite discrete G-module M killed by pⁱ, so that the pairings

Hʲ(G, M) × H²⁻ʲ(G, Hom(M, I(χ)/pⁱ)) → H²(G, I(χ)/pⁱ) ≅ ℤ/pⁱ

are perfect pairings of finite abelian groups. The argument has three steps.

• Modules killed by p. Multiplication by pⁱ⁻¹ is an injective G-map I(χ)/p →+[G] I(χ)/pⁱ whose range is the p-torsion of I(χ)/pⁱ, and the induced map H²(G, I(χ)/p) → H²(G, I(χ)/pⁱ) is injective, by the prescription property of χ, with range the p-torsion of the cyclic group H²(G, I(χ)/pⁱ), by counting (TauCeti.IsDemushkin.exists_explicitCoeff2_mulPow_eq_of_nsmul_eq_zero). On a module M killed by p, the internal homs Hom(M, I(χ)/p) and Hom(M, I(χ)/pⁱ) are therefore identified, and every homomorphism from the p-torsion group H²⁻ʲ(G, Hom(M, I(χ)/p)) to H²(G, I(χ)/pⁱ) factors through H²(G, I(χ)/p); so each αⱼ at level pⁱ is the one at level p read through these identifications (TauCeti.ContCohomology.dualityMap0_bijective_of_injective_of_forall_nsmul_eq_zero and its companions), and I(χ)/p is 𝔽_p with trivial action, where the duality is TauCeti.IsDemushkin.dualityMap0_bijective, dualityMap1_bijective, dualityMap2_bijective.
• Dévissage. Tate's dévissage over the order of M for an infinite pro-p group (TauCeti.IsProP.dualityMap0_bijective_dualityMap1_bijective_dualityMap2_injective), whose base case is the first step on the trivial modules of order p, applies because both I(χ)/pⁱ and H²(G, I(χ)/pⁱ) ≅ ℤ/pⁱ satisfy Baer's criterion over ℤ/pⁱ. It gives α₀ and α₁ bijective and α₂ injective.
• Counting. α₂ is then bijective by the general count TauCeti.ContCohomology.dualityMap2_bijective_of_injective_of_addEquiv_zmod, as both I(χ)/pⁱ and H²(G, I(χ)/pⁱ) are ℤ/pⁱ: |H²(G, M)| = |H⁰(G, Hom(M, I(χ)/pⁱ))|, since α₀ is bijective on the dual module and M is its own double dual with values in I(χ)/pⁱ.

Main results

• TauCeti.IsDemushkin.dualityMap0_zModTwist_bijective, TauCeti.IsDemushkin.dualityMap1_zModTwist_bijective, TauCeti.IsDemushkin.dualityMap2_zModTwist_bijective: Tate's duality at level pⁱ: for an infinite Demushkin group and a finite discrete G-module M killed by pⁱ, Tate's duality maps with coefficients I(χ)/pⁱ are bijective.

References

• J.-P. Serre, Structure de certains pro-p-groupes (d'après Demuškin), Séminaire Bourbaki 8 (1962/63), exposé 252, §9.
• J. P. Labute, Classification of Demushkin groups, Canad. J. Math. 19 (1967), 106–132, §2.

Modules killed by p: reduction to the duality at 𝔽_p

Dévissage over the order of M and the duality at level pⁱ

theorem TauCeti.IsDemushkin.dualityMap0_zModTwist_bijective {p : ℕ} [Fact (Nat.Prime p)] {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] [Infinite G] (hG : IsDemushkin p G) (i : ℕ) (M : Type u) [AddCommGroup M] [TopologicalSpace M] [DiscreteTopology M] [DistribMulAction G M] [ContinuousSMul G M] [Finite M] (hM : ∀ (x : M), p ^ i • x = 0) : Function.Bijective ⇑(ContCohomology.dualityMap0 G M (ZModTwist (demushkinCharacter hG) i))

Tate's duality map α₀ of an infinite Demushkin group with coefficients I(χ)/pⁱ is bijective on every finite discrete G-module M killed by pⁱ: H⁰(G, M) × H²(G, Hom(M, I(χ)/pⁱ)) → H²(G, I(χ)/pⁱ) is a perfect pairing. Injectivity is the general TauCeti.IsProP.dualityMap0_injective, from the co-effaceability of H⁰.

theorem TauCeti.IsDemushkin.dualityMap1_zModTwist_bijective {p : ℕ} [Fact (Nat.Prime p)] {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] [Infinite G] (hG : IsDemushkin p G) (i : ℕ) (M : Type u) [AddCommGroup M] [TopologicalSpace M] [DiscreteTopology M] [DistribMulAction G M] [ContinuousSMul G M] [Finite M] (hM : ∀ (x : M), p ^ i • x = 0) : Function.Bijective ⇑(ContCohomology.dualityMap1 G M (ZModTwist (demushkinCharacter hG) i))

Tate's duality map α₁ of an infinite Demushkin group with coefficients I(χ)/pⁱ is bijective on every finite discrete G-module M killed by pⁱ: H¹(G, M) × H¹(G, Hom(M, I(χ)/pⁱ)) → H²(G, I(χ)/pⁱ) is a perfect pairing.

theorem TauCeti.IsDemushkin.dualityMap2_zModTwist_bijective {p : ℕ} [Fact (Nat.Prime p)] {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] [Infinite G] (hG : IsDemushkin p G) (i : ℕ) (M : Type u) [AddCommGroup M] [TopologicalSpace M] [DiscreteTopology M] [DistribMulAction G M] [ContinuousSMul G M] [Finite M] (hM : ∀ (x : M), p ^ i • x = 0) : Function.Bijective ⇑(ContCohomology.dualityMap2 G M (ZModTwist (demushkinCharacter hG) i))

Tate's duality map α₂ of an infinite Demushkin group with coefficients I(χ)/pⁱ is bijective on every finite discrete G-module M killed by pⁱ: H²(G, M) × H⁰(G, Hom(M, I(χ)/pⁱ)) → H²(G, I(χ)/pⁱ) is a perfect pairing. It is injective by the dévissage and bijective by counting (TauCeti.ContCohomology.dualityMap2_bijective_of_injective_of_addEquiv_zmod), since α₀ is bijective on Hom(M, I(χ)/pⁱ) and both I(χ)/pⁱ and H²(G, I(χ)/pⁱ) are ℤ/pⁱ.