Cohomological dimension and subgroups of index prime to p

Let G be a profinite group and p a prime. Restriction to an open subgroup U of index prime to p is injective on the cohomology of every discrete p-primary torsion G-module, in every degree. In degree zero this is the inclusion of invariants X^G ⊆ X^U (TauCeti.ContinuousCohomology.res_zero_injective); in positive degree a class killed by restriction to U is killed by [G : U] (TauCeti.ContinuousCohomology.index_nsmul_eq_zero_of_res_eq_zero), and it is also killed by a power of p, so it is zero. The same holds for a closed subgroup H all of whose open neighbourhoods V ⊇ H have index prime to p: a class restricting to zero on H already restricts to zero on some open V ⊇ H, by the colimit description of the cohomology of a closed subgroup (TauCeti.ContinuousCohomology.exists_openSubgroup_le_res_eq_zero).

Consequently cd_p G ≤ cd_p U for an open subgroup U of index prime to p of a compact group, and cd_p G ≤ cd_p H for such a closed subgroup H of a profinite group: both follow from the one observation that vanishing transfers from a subgroup on which restriction is injective in every positive degree (TauCeti.CohomologicalDimensionLE.of_forall_res_injective). Combined with closed-subgroup monotonicity from Shapiro's lemma, this gives cd_p G = cd_p H for such a closed subgroup, and in particular for an open subgroup of index prime to p. These equalities hold in ℕ∞, including infinite cohomological dimension. The closed-subgroup statement is the one a Sylow pro-p subgroup satisfies, since each of its open neighbourhoods has index prime to p; the comparison of cd_p G with the cohomological dimension of a Sylow pro-p subgroup is where these results are used.

Main results

• TauCeti.ContinuousCohomology.res_injective_of_not_dvd_index: for a compact group G, an open subgroup U of index prime to p and a discrete p-primary torsion representation X, restriction Hⁿ(G, X) → Hⁿ(U, X) is injective in every degree n.
• TauCeti.ContinuousCohomology.res_injective_of_forall_not_dvd_index: the same for a closed subgroup H of a profinite group all of whose open neighbourhoods have index prime to p, for smooth discrete p-primary torsion X.
• TauCeti.cohomologicalDimensionAt_le_of_not_dvd_index: cd_p G ≤ cd_p U for an open subgroup U of a compact group G with [G : U] prime to p.
• TauCeti.cohomologicalDimensionAt_le_of_isClosed_of_forall_not_dvd_index: cd_p G ≤ cd_p H for such a closed subgroup H of a profinite group G.
• TauCeti.cohomologicalDimensionAt_eq_of_isClosed_of_forall_not_dvd_index: cd_p G = cd_p H for such a closed subgroup, with no finiteness assumption on cohomological dimension.
• TauCeti.cohomologicalDimensionAt_eq_of_not_dvd_index: cd_p G = cd_p U for an open subgroup of index prime to p in a profinite group.

References

• J.-P. Serre, Galois Cohomology, Ch. I, §3.3, Prop. 14 and its Corollary.
• J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields, 2nd ed., (3.3.5) and (3.3.6).

theorem TauCeti.ContinuousCohomology.res_injective_of_not_dvd_index {p : ℕ} {k : Type u} {G : Type v} [Ring k] [TopologicalSpace k] [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] {X : TopRep k G} [DiscreteTopology ↑X] (hp : Nat.Prime p) (hX : IsAddPGroup p ↑X) (U : OpenSubgroup G) (hU : ¬p ∣ (↑U).index) (n : ℕ) : Function.Injective ⇑(TopModuleCat.Hom.hom (res (↑U) X n))

Restriction to an open subgroup of index prime to p is injective on the cohomology of a discrete p-primary torsion representation X of a compact group G, in every degree.

theorem TauCeti.ContinuousCohomology.res_injective_of_forall_not_dvd_index {p : ℕ} {k : Type u} {G : Type v} [Ring k] [TopologicalSpace k] [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] {X : TopRep k G} [TotallyDisconnectedSpace G] (hp : Nat.Prime p) (hX : IsSmoothDiscrete k X) (hX' : IsAddPGroup p ↑X) {H : Subgroup G} (hH : IsClosed ↑H) (hind : ∀ (V : OpenSubgroup G), H ≤ ↑V → ¬p ∣ (↑V).index) (n : ℕ) : Function.Injective ⇑(TopModuleCat.Hom.hom (res H X n))

Restriction to a closed subgroup whose open neighbourhoods all have index prime to p is injective on the cohomology of a smooth discrete p-primary torsion representation X of a profinite group G, in every degree.

theorem TauCeti.CohomologicalDimensionLE.of_not_dvd_index {p : ℕ} {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] (hp : Nat.Prime p) (U : OpenSubgroup G) (hU : ¬p ∣ (↑U).index) {n : ℕ} (h : CohomologicalDimensionLE p (↥↑U) n) : CohomologicalDimensionLE p G n

cd_p G ≤ cd_p U for an open subgroup U of index prime to p, as the vanishing predicate: for U open in a compact group G with ¬ p ∣ [G : U], CohomologicalDimensionLE p U n implies CohomologicalDimensionLE p G n.

theorem TauCeti.cohomologicalDimensionAt_le_of_not_dvd_index {p : ℕ} {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] (hp : Nat.Prime p) (U : OpenSubgroup G) (hU : ¬p ∣ (↑U).index) : cohomologicalDimensionAt p G ≤ cohomologicalDimensionAt p ↥↑U

cd_p G ≤ cd_p U for an open subgroup U of index prime to p (Serre, Galois Cohomology, I §3.3, Prop. 14): for U open in a compact group G with ¬ p ∣ [G : U], the p-cohomological dimension of G is at most that of U.

theorem TauCeti.CohomologicalDimensionLE.of_isClosed_of_forall_not_dvd_index {p : ℕ} {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] (hp : Nat.Prime p) {H : Subgroup G} (hH : IsClosed ↑H) (hind : ∀ (V : OpenSubgroup G), H ≤ ↑V → ¬p ∣ (↑V).index) {n : ℕ} (h : CohomologicalDimensionLE p (↥H) n) : CohomologicalDimensionLE p G n

cd_p G ≤ cd_p H for a closed subgroup H whose open neighbourhoods all have index prime to p, as the vanishing predicate: for H closed in a profinite group G with every open V ⊇ H of index prime to p, CohomologicalDimensionLE p H n implies CohomologicalDimensionLE p G n.

theorem TauCeti.cohomologicalDimensionAt_le_of_isClosed_of_forall_not_dvd_index {p : ℕ} {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] (hp : Nat.Prime p) {H : Subgroup G} (hH : IsClosed ↑H) (hind : ∀ (V : OpenSubgroup G), H ≤ ↑V → ¬p ∣ (↑V).index) : cohomologicalDimensionAt p G ≤ cohomologicalDimensionAt p ↥H

cd_p G ≤ cd_p H for a closed subgroup H whose open neighbourhoods all have index prime to p (Serre, Galois Cohomology, I §3.3, Cor. to Prop. 14): for H closed in a profinite group G with every open V ⊇ H of index prime to p, the p-cohomological dimension of G is at most that of H.

theorem TauCeti.cohomologicalDimensionAt_eq_of_isClosed_of_forall_not_dvd_index {p : ℕ} {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] (hp : Nat.Prime p) {H : Subgroup G} (hH : IsClosed ↑H) (hind : ∀ (V : OpenSubgroup G), H ≤ ↑V → ¬p ∣ (↑V).index) : cohomologicalDimensionAt p G = cohomologicalDimensionAt p ↥H

cd_p G = cd_p H for a closed subgroup whose open neighbourhoods have index prime to p. The equality is in ℕ∞ and does not assume that either cohomological dimension is finite.

theorem TauCeti.cohomologicalDimensionAt_eq_of_not_dvd_index {p : ℕ} {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] (hp : Nat.Prime p) (U : OpenSubgroup G) (hU : ¬p ∣ (↑U).index) : cohomologicalDimensionAt p G = cohomologicalDimensionAt p ↥↑U

An open subgroup of index prime to p has the same p-cohomological dimension as the ambient profinite group (NSW (3.3.5)). Unlike the equality for arbitrary open subgroups, this holds even when the cohomological dimension is infinite.