Finiteness of the cohomology of a pro-p group with finite coefficients

Let G be a compact pro-p group and n a degree. If Hⁿ(G, 𝔽_p) is finite, then Hⁿ(G, M) is finite for every finite discrete p-primary G-module M, whatever the action. This is the finiteness dévissage TauCeti.IsProP.finite_continuousCohomology_of_forall_natCard_eq_smul_eq_self with 𝔽_p as the test module: a trivial discrete G-module of order p has the cohomology of 𝔽_p (TauCeti.finite_continuousCohomology_iff_finite_cohomFp_of_natCard_eq).

In degree 2 the statement is restated on the explicit cocycle model H2 G M. For a profinite pro-p group, through the permutation module Coind_U^G 𝔽_p of an open subgroup U and Shapiro's lemma in every degree, it gives that Hⁿ(U, 𝔽_p) is finite whenever Hⁿ(G, 𝔽_p) is. No bound on the cohomological dimension of G enters: finiteness of Hⁿ(G, 𝔽_p) alone controls the finite coefficients in degree n. For a finitely presented pro-p group H²(G, 𝔽_p) is finite, so H²(G, M) is finite for every finite p-primary M; the case of a Demushkin group, where H²(G, 𝔽_p) is one-dimensional, is the one Tate's duality argument uses.

Main results

• TauCeti.IsProP.finite_continuousCohomology_of_finite_cohomFp: finiteness of Hⁿ on finite p-primary modules when Hⁿ(G, 𝔽_p) is finite.
• TauCeti.IsProP.finite_H2: the degree-2 case on the explicit H2 G M.
• TauCeti.IsProP.finite_cohomFp_openSubgroup: Hⁿ(U, 𝔽_p) is finite for every open subgroup U when Hⁿ(G, 𝔽_p) is.

References

• J.-P. Serre, Galois Cohomology, Ch. I, §4.1.
• J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields, 2nd ed., (3.3.2).

theorem TauCeti.IsProP.finite_continuousCohomology_of_finite_cohomFp {p : ℕ} [hp : Fact (Nat.Prime p)] {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] (hG : IsProP p G) {n : ℕ} [Finite ↑(cohomFp p G n).toModuleCat] (M : Type u) [AddCommGroup M] [TopologicalSpace M] [DiscreteTopology M] [DistribMulAction G M] [ContinuousSMul G M] [Finite M] (hM : IsAddPGroup p M) : Finite ↑(continuousCohomology n (ofDiscreteModule ℤ G M)).toModuleCat

Finiteness of Hⁿ on finite p-primary modules. Let G be a compact pro-p group with Hⁿ(G, 𝔽_p) finite. Then Hⁿ(G, M) is finite for every finite discrete p-primary G-module M.

theorem TauCeti.IsProP.finite_H2 {p : ℕ} [hp : Fact (Nat.Prime p)] {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] (hG : IsProP p G) [Finite ↑(cohomFp p G 2).toModuleCat] (M : Type u) [AddCommGroup M] [TopologicalSpace M] [DiscreteTopology M] [DistribMulAction G M] [ContinuousSMul G M] [Finite M] (hM : IsAddPGroup p M) : Finite (ContCohomology.H2 G M)

Finiteness of H² on finite p-primary modules. Let G be a compact pro-p group with H²(G, 𝔽_p) finite. Then the explicit H²(G, M) is finite for every finite discrete p-primary G-module M.

theorem TauCeti.IsProP.finite_cohomFp_openSubgroup {p : ℕ} [hp : Fact (Nat.Prime p)] {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] (hG : IsProP p G) {n : ℕ} [Finite ↑(cohomFp p G n).toModuleCat] (U : OpenSubgroup G) : Finite ↑(cohomFp p (↥↑U) n).toModuleCat

Finiteness of Hⁿ(-, 𝔽_p) on open subgroups. Let G be a profinite pro-p group with Hⁿ(G, 𝔽_p) finite. Then Hⁿ(U, 𝔽_p) is finite for every open subgroup U: it is Hⁿ(G, Coind_U^G 𝔽_p) by Shapiro's lemma, and Coind_U^G 𝔽_p is a finite p-primary module.