The second cohomology 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. Then H²(G, I(χ)/pⁱ) is cyclic of order pⁱ, so that H²(G, I(χ)/pⁱ) ≅ ℤ/pⁱ. This is the finite-level form of the statement that the dualizing module of G is ℚ_p/ℤ_p with G acting through its orientation (Serre's exposé, §9), and it is the target group of the perfect pairings Hʲ(G, M) × H²⁻ʲ(G, Hom(M, I(χ)/pⁱ)) → H²(G, I(χ)/pⁱ).

The proof assembles three facts: the canonical character has the prescription property, so multiplication by p is injective on H²(G, I(χ)/pⁱ) → H²(G, I(χ)/pⁱ⁺¹); cd_p G = 2 (TauCeti.IsDemushkin.cohomologicalDimensionAt_eq_two), so the reductions on H² are surjective; and H²(G, I(χ)/p) = H²(G, 𝔽_p) has order p, because the action of a pro-p group on I(χ)/p is trivial. The general argument is TauCeti.HasPrescriptionProperty.isAddCyclic_H2_zModTwist and its companions.

Main results

• TauCeti.IsDemushkin.natCard_H2_zModTwist_demushkinCharacter_one: H²(G, I(χ)/p) has order p, for every Demushkin group.
• TauCeti.IsDemushkin.surjective_explicitCoeff2_reduce_demushkinCharacter: for an infinite Demushkin group, every reduction H²(G, I(χ)/pⁿ) → H²(G, I(χ)/pʲ) is surjective.
• TauCeti.IsDemushkin.natCard_H2_zModTwist_demushkinCharacter, TauCeti.IsDemushkin.isAddCyclic_H2_zModTwist_demushkinCharacter, TauCeti.IsDemushkin.nonempty_addEquiv_H2_zModTwist_demushkinCharacter_zmod: for an infinite Demushkin group, H²(G, I(χ)/pⁱ) is cyclic of order pⁱ, isomorphic to ℤ/pⁱ.
• TauCeti.IsDemushkin.moduleBaer_H2_zModTwist_demushkinCharacter: for an infinite Demushkin group, H²(G, I(χ)/pⁱ) satisfies Baer's criterion over ℤ/pⁱ.
• TauCeti.IsDemushkin.exists_explicitCoeff2_mulPow_eq_of_nsmul_eq_zero: for an infinite Demushkin group, the pⁱ-torsion of H²(G, I(χ)/pⁱ⁺ʲ) is the image of H²(G, I(χ)/pⁱ) under multiplication by pʲ.

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.

theorem TauCeti.IsDemushkin.natCard_H2_zModTwist_demushkinCharacter_one {p : ℕ} [Fact (Nat.Prime p)] {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] (hG : IsDemushkin p G) : Nat.card (ContCohomology.H2 G (ZModTwist (demushkinCharacter hG) 1)) = p

H²(G, I(χ)/p) has order p for a Demushkin group G with canonical character χ: a pro-p group acts trivially on the bottom level I(χ)/p, which is therefore 𝔽_p, and H²(G, 𝔽_p) is one-dimensional.

theorem TauCeti.IsDemushkin.surjective_explicitCoeff2_reduce_demushkinCharacter {p : ℕ} [Fact (Nat.Prime p)] {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] (hG : IsDemushkin p G) [Infinite G] {j n : ℕ} (hj : j ≤ n) : Function.Surjective ⇑(ContCohomology.explicitCoeff2 G (ZModTwist (demushkinCharacter hG) n) (ZModTwist.reduce (demushkinCharacter hG) hj) ⋯)

H² is right exact on the twisted coefficients of an infinite Demushkin group: every reduction H²(G, I(χ)/pⁿ) → H²(G, I(χ)/pʲ), j ≤ n, is surjective, because cd_p G = 2.

theorem TauCeti.IsDemushkin.natCard_H2_zModTwist_demushkinCharacter {p : ℕ} [Fact (Nat.Prime p)] {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] (hG : IsDemushkin p G) [Infinite G] (i : ℕ) : Nat.card (ContCohomology.H2 G (ZModTwist (demushkinCharacter hG) i)) = p ^ i

The order of H²(G, I(χ)/pⁱ) for an infinite Demushkin group G with canonical character χ: it is pⁱ.

theorem TauCeti.IsDemushkin.isAddCyclic_H2_zModTwist_demushkinCharacter {p : ℕ} [Fact (Nat.Prime p)] {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] (hG : IsDemushkin p G) [Infinite G] (i : ℕ) : IsAddCyclic (ContCohomology.H2 G (ZModTwist (demushkinCharacter hG) i))

H²(G, I(χ)/pⁱ) is cyclic for an infinite Demushkin group G with canonical character χ.

theorem TauCeti.IsDemushkin.nonempty_addEquiv_H2_zModTwist_demushkinCharacter_zmod {p : ℕ} [Fact (Nat.Prime p)] {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] (hG : IsDemushkin p G) [Infinite G] (i : ℕ) : Nonempty (ContCohomology.H2 G (ZModTwist (demushkinCharacter hG) i) ≃+ ZMod (p ^ i))

H²(G, I(χ)/pⁱ) ≅ ℤ/pⁱ for an infinite Demushkin group G with canonical character χ: the top cohomology of the twisted coefficients at level i is cyclic of order pⁱ.

theorem TauCeti.IsDemushkin.moduleBaer_H2_zModTwist_demushkinCharacter {p : ℕ} [Fact (Nat.Prime p)] {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] (hG : IsDemushkin p G) [Infinite G] (i : ℕ) : Module.Baer (ZMod (p ^ i)) (ContCohomology.H2 G (ZModTwist (demushkinCharacter hG) i))

H²(G, I(χ)/pⁱ) is self-injective over ℤ/pⁱ for an infinite Demushkin group G with canonical character χ: being ℤ/pⁱ, it satisfies Baer's criterion over ℤ/pⁱ, so that Hom(-, H²(G, I(χ)/pⁱ)) is exact on the groups killed by pⁱ.

theorem TauCeti.IsDemushkin.exists_explicitCoeff2_mulPow_eq_of_nsmul_eq_zero {p : ℕ} [Fact (Nat.Prime p)] {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] (hG : IsDemushkin p G) [Infinite G] {i j n : ℕ} (h : i + j = n) {y : ContCohomology.H2 G (ZModTwist (demushkinCharacter hG) n)} (hy : p ^ i • y = 0) : ∃ (x : ContCohomology.H2 G (ZModTwist (demushkinCharacter hG) i)), (ContCohomology.explicitCoeff2 G (ZModTwist (demushkinCharacter hG) i) (ZModTwist.mulPow (demushkinCharacter hG) h) ⋯) x = y

The pⁱ-torsion of H²(G, I(χ)/pⁿ) is the image of H²(G, I(χ)/pⁱ), for i + j = n and an infinite Demushkin group G with canonical character χ: multiplication by pʲ is injective on H² by the prescription property, its image lies in the pⁱ-torsion, and the pⁱ-torsion of the cyclic group H²(G, I(χ)/pⁿ) has at most pⁱ elements.