Cohomological dimension and subgroups with injective restriction

Let G be a topological group and H a subgroup. If restriction Hⁱ⁺¹(G, M) → Hⁱ⁺¹(H, M) is injective for every discrete p-primary torsion G-module M and every i, then vanishing of cohomology transfers from H to G: CohomologicalDimensionLE p H n implies CohomologicalDimensionLE p G n, and consequently cd_p G ≤ cd_p H. No topological hypothesis on G or H is needed.

This is the common formal step behind the comparisons of cd_p G with the cohomological dimension of a subgroup on which restriction is injective, such as an open subgroup of index prime to p or a closed subgroup all of whose open neighbourhoods have index prime to p (TauCeti.RepresentationTheory.Homological.ContCohomology.CohomologicalDimension.IndexNotDvd).

Main results

• TauCeti.CohomologicalDimensionLE.of_forall_res_injective: vanishing transfers along a subgroup on which restriction is injective in every positive degree.
• TauCeti.cohomologicalDimensionAt_le_of_forall_res_injective: cd_p G ≤ cd_p H for a subgroup H on which restriction is injective in every positive degree, for every discrete p-primary torsion G-module.

References

• J.-P. Serre, Galois Cohomology, Ch. I, §3.3.
• J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields, 2nd ed., Ch. III §3.

theorem TauCeti.CohomologicalDimensionLE.of_forall_res_injective {p : ℕ} {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] {H : Subgroup G} (hres : ∀ (M : Type (max u v)) [inst : AddCommGroup M] [inst_1 : TopologicalSpace M] [inst_2 : DiscreteTopology M] [inst_3 : DistribMulAction G M] [ContinuousSMul G M], IsAddPGroup p M → ∀ (i : ℕ), Function.Injective ⇑(TopModuleCat.Hom.hom (ContinuousCohomology.res H (ofDiscreteModule ℤ G M) (i + 1)))) {n : ℕ} (h : CohomologicalDimensionLE p (↥H) n) : CohomologicalDimensionLE p G n

Vanishing transfers along a subgroup on which restriction is injective, as the vanishing predicate: if restriction Hⁱ⁺¹(G, M) → Hⁱ⁺¹(H, M) is injective for every discrete p-primary torsion G-module M and every i, then CohomologicalDimensionLE p H n implies CohomologicalDimensionLE p G n.

theorem TauCeti.cohomologicalDimensionAt_le_of_forall_res_injective {p : ℕ} {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] {H : Subgroup G} (hres : ∀ (M : Type (max u v)) [inst : AddCommGroup M] [inst_1 : TopologicalSpace M] [inst_2 : DiscreteTopology M] [inst_3 : DistribMulAction G M] [ContinuousSMul G M], IsAddPGroup p M → ∀ (i : ℕ), Function.Injective ⇑(TopModuleCat.Hom.hom (ContinuousCohomology.res H (ofDiscreteModule ℤ G M) (i + 1)))) : cohomologicalDimensionAt p G ≤ cohomologicalDimensionAt p ↥H

cd_p G ≤ cd_p H for a subgroup H on which restriction is injective: if restriction Hⁱ⁺¹(G, M) → Hⁱ⁺¹(H, M) is injective for every discrete p-primary torsion G-module M and every i, then the p-cohomological dimension of G is at most that of H.