Finiteness of continuous cohomology with finite coefficients, from an open normal subgroup

Let G be a profinite group, V an open normal subgroup of G and N a natural number. This file reduces the finiteness of the continuous cohomology Hⁱ(G, A) of the finite discrete G-modules A killed by N on which V acts trivially to the finiteness of Hʲ(V, M), for 0 < j ≤ i, on the trivial V-modules M of prime order dividing N. Over a field this is the step from an absolute Galois group to that of a finite Galois extension over which the coefficients become constant and which contains the relevant roots of unity; this is how the finiteness of the local Galois cohomology of a finite module is proved.

There are two steps.

• Coinduction. The unit A ↪ Coind_V^G A of coinduction embeds A into a finite module with Hʲ(G, Coind_V^G A) ≅ Hʲ(V, A) by Shapiro's lemma, and the cokernel C of the short exact sequence TauCeti.ContCohomology.coindShortExact is again a finite module killed by N on which V acts trivially, because V is normal (TauCeti.ContCohomology.CoindQuotient.smul_eq_self_of_forall_smul_eq_self). Exactness of Hⁱ(G, C) → Hⁱ⁺¹(G, A) → Hⁱ⁺¹(G, Coind_V^G A) then gives finiteness of Hⁱ⁺¹(G, A) by induction on the degree, starting from H⁰(G, A) ⊆ A (TauCeti.ContCohomology.finite_continuousCohomology_zero, TauCeti.ContinuousCohomology.finite_continuousCohomology_of_isOpen_of_normal).
• Dévissage of trivial modules. A finite trivial module killed by N is an iterated extension of trivial modules of prime order dividing N, by Cauchy's theorem and the dévissage induction principle TauCeti.ContCohomology.finite_induction_of_exists_addSubgroup; exactness of Hʲ(V, P) → Hʲ(V, M) → Hʲ(V, M ⧸ P) passes finiteness along each extension (TauCeti.ContinuousCohomology.finite_continuousCohomology_of_forall_natCard_prime).

Main results

• TauCeti.ContinuousCohomology.finite_continuousCohomology_of_forall_natCard_prime: finiteness of Hⁿ on the trivial modules of prime order dividing N gives finiteness on every finite trivial module killed by N.
• TauCeti.ContinuousCohomology.finite_continuousCohomology_of_isOpen_of_normal: finiteness of Hʲ(V, -), 0 < j ≤ i, on the finite trivial modules killed by N gives finiteness of Hⁱ(G, A) for every finite discrete A killed by N on which V acts trivially.
• TauCeti.ContinuousCohomology.finite_continuousCohomology_of_isOpen_of_normal_of_prime: the same, with the trivial modules of prime order dividing N as test class.

References

• J.-P. Serre, Galois Cohomology, Ch. II, §5.2, proof of Proposition 14.
• J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields, 2nd ed., (1.6.4) for Shapiro's lemma and Ch. VII, §1 for the local finiteness theorem.

theorem TauCeti.ContinuousCohomology.finite_continuousCohomology_of_forall_natCard_prime {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [LocallyCompactSpace G] {N n : ℕ} (h : ∀ (A : Type u) [inst : AddCommGroup A] [inst_1 : TopologicalSpace A] [inst_2 : DiscreteTopology A] [inst_3 : DistribMulAction G A] [ContinuousSMul G A] [Finite A], Nat.Prime (Nat.card A) → Nat.card A ∣ N → (∀ (g : G) (a : A), g • a = a) → Finite ↑(continuousCohomology n (ofDiscreteModule ℤ G A)).toModuleCat) (M : Type u) [AddCommGroup M] [TopologicalSpace M] [DiscreteTopology M] [DistribMulAction G M] [ContinuousSMul G M] [Finite M] (hN : ∀ (m : M), N • m = 0) (htriv : ∀ (g : G) (m : M), g • m = m) : Finite ↑(continuousCohomology n (ofDiscreteModule ℤ G M)).toModuleCat

Dévissage of finite trivial coefficients. Let G be a locally compact group, n a degree and N a natural number. If Hⁿ(G, A) is finite for every discrete G-module A of prime order dividing N on which G acts trivially, then Hⁿ(G, M) is finite for every finite discrete G-module M killed by N on which G acts trivially.

theorem TauCeti.ContinuousCohomology.finite_continuousCohomology_of_isOpen_of_normal {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] {V : Subgroup G} [V.Normal] (hV : IsOpen ↑V) {N : ℕ} (i : ℕ) (h : ∀ (j : ℕ), 0 < j → j ≤ i → ∀ (M : Type u) [inst : AddCommGroup M] [inst_1 : TopologicalSpace M] [inst_2 : DiscreteTopology M] [inst_3 : DistribMulAction (↥V) M] [ContinuousSMul (↥V) M] [Finite M], (∀ (m : M), N • m = 0) → (∀ (v : ↥V) (m : M), v • m = m) → Finite ↑(continuousCohomology j (ofDiscreteModule ℤ (↥V) M)).toModuleCat) (A : Type u) [AddCommGroup A] [TopologicalSpace A] [DiscreteTopology A] [DistribMulAction G A] [ContinuousSMul G A] [Finite A] (hN : ∀ (a : A), N • a = 0) (htriv : ∀ v ∈ V, ∀ (a : A), v • a = a) : Finite ↑(continuousCohomology i (ofDiscreteModule ℤ G A)).toModuleCat

Finiteness through an open normal subgroup. Let G be a profinite group, V an open normal subgroup, N a natural number and i a degree. Suppose that for every 0 < j ≤ i, Hʲ(V, M) is finite for every finite discrete V-module M killed by N on which V acts trivially. Then Hⁱ(G, A) is finite for every finite discrete G-module A killed by N on which V acts trivially. Degree zero needs no hypothesis, H⁰(G, A) being a subgroup of A.

theorem TauCeti.ContinuousCohomology.finite_continuousCohomology_of_isOpen_of_normal_of_prime {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] {V : Subgroup G} [V.Normal] (hV : IsOpen ↑V) {N : ℕ} (i : ℕ) (h : ∀ (j : ℕ), 0 < j → j ≤ i → ∀ (M : Type u) [inst : AddCommGroup M] [inst_1 : TopologicalSpace M] [inst_2 : DiscreteTopology M] [inst_3 : DistribMulAction (↥V) M] [ContinuousSMul (↥V) M] [Finite M], Nat.Prime (Nat.card M) → Nat.card M ∣ N → (∀ (v : ↥V) (m : M), v • m = m) → Finite ↑(continuousCohomology j (ofDiscreteModule ℤ (↥V) M)).toModuleCat) (A : Type u) [AddCommGroup A] [TopologicalSpace A] [DiscreteTopology A] [DistribMulAction G A] [ContinuousSMul G A] [Finite A] (hN : ∀ (a : A), N • a = 0) (htriv : ∀ v ∈ V, ∀ (a : A), v • a = a) : Finite ↑(continuousCohomology i (ofDiscreteModule ℤ G A)).toModuleCat

Finiteness through an open normal subgroup, prime test class. Let G be a profinite group, V an open normal subgroup, N a natural number and i a degree. Suppose that for every 0 < j ≤ i, Hʲ(V, M) is finite for every discrete V-module M of prime order dividing N on which V acts trivially. Then Hⁱ(G, A) is finite for every finite discrete G-module A killed by N on which V acts trivially.