The prescription property from Tate's duality map

Let G be a topological group, χ : G →ₜ* ℤ_pˣ a continuous character and I(χ)/pⁱ its twisted coefficients. Tate's duality map in degree two with coefficients I(χ)/pⁿ, α₂ : H²(G, M) → Hom(H⁰(G, Hom(M, I(χ)/pⁿ)), H²(G, I(χ)/pⁿ)), is the adjoint of the evaluation pairing (TauCeti.ContCohomology.dualityMap2). Its injectivity on the twist M = I(χ)/pⁱ itself, for i + j = n, forces multiplication by pʲ to be injective on H²(G, I(χ)/pⁱ) → H²(G, I(χ)/pⁿ): a class killed by the multiplication pairs to zero with every invariant endomorphism of I(χ)/pⁿ restricted along it, and every invariant homomorphism I(χ)/pⁱ → I(χ)/pⁿ is such a restriction (TauCeti.ZModTwist.surjective_explicitCoeff0_precomp_mulPow). Injectivity of multiplication by p on H² between consecutive levels is Labute's cohomological form of the prescription property (TauCeti.hasPrescriptionProperty_iff_forall_injective_explicitCoeff2_mulPow), so a character whose duality maps α₂ are injective between consecutive levels has the prescription property.

This is the mechanism by which a duality theorem identifies the canonical character of a Demushkin group, which is the unique character with the prescription property: a duality with values in H²(G, I(χ)/pⁿ) that is perfect, or merely injective in degree two, on the twists of χ can only hold for the canonical character.

Main results

• TauCeti.ZModTwist.injective_explicitCoeff2_mulPow_of_injective_dualityMap2: if α₂ with values in H²(G, I(χ)/pⁿ) is injective on I(χ)/pⁱ, i + j = n, then multiplication by pʲ is injective on H²(G, I(χ)/pⁱ) → H²(G, I(χ)/pⁿ).
• TauCeti.hasPrescriptionProperty_of_forall_injective_dualityMap2: a character whose duality maps α₂ are injective between consecutive levels has the prescription property.

References

• J. P. Labute, Classification of Demushkin groups, Canad. J. Math. 19 (1967), 106–132, §2, Proposition 6.
• J.-P. Serre, Structure de certains pro-p-groupes (d'après Demuškin), Séminaire Bourbaki 8 (1962/63), exposé 252, §9.

theorem TauCeti.ZModTwist.injective_explicitCoeff2_mulPow_of_injective_dualityMap2 {p : ℕ} [Fact (Nat.Prime p)] {G : Type u} [Group G] [TopologicalSpace G] [ContinuousMul G] (χ : G →ₜ* ℤ_[p]ˣ) {i j n : ℕ} (h : i + j = n) (hα : Function.Injective ⇑(ContCohomology.dualityMap2 G (ZModTwist χ i) (ZModTwist χ n))) : Function.Injective ⇑(ContCohomology.explicitCoeff2 G (ZModTwist χ i) (mulPow χ h) ⋯)

Injectivity of α₂ on a twist gives injectivity of the multiplication on H². If Tate's duality map α₂ : H²(G, I(χ)/pⁱ) → Hom(H⁰(G, Hom(I(χ)/pⁱ, I(χ)/pⁿ)), H²(G, I(χ)/pⁿ)) is injective, i + j = n, then multiplication by pʲ is injective on H²(G, I(χ)/pⁱ) → H²(G, I(χ)/pⁿ).

theorem TauCeti.hasPrescriptionProperty_of_forall_injective_dualityMap2 {p : ℕ} [Fact (Nat.Prime p)] {G : Type u} [Group G] [TopologicalSpace G] [ContinuousMul G] (χ : G →ₜ* ℤ_[p]ˣ) (h : ∀ (i : ℕ), Function.Injective ⇑(ContCohomology.dualityMap2 G (ZModTwist χ i) (ZModTwist χ (i + 1)))) : HasPrescriptionProperty χ

A character whose duality maps α₂ are injective between consecutive levels has the prescription property: if, for every i, Tate's duality map with values in H²(G, I(χ)/pⁱ⁺¹) is injective on I(χ)/pⁱ, then every reduction H¹(G, I(χ)/pⁱ) → H¹(G, I(χ)/p) is surjective.