Cohomological dimension passes to closed subgroups

Let G be a profinite group and U a closed subgroup. Then cd_p U ≤ cd_p G and scd_p U ≤ scd_p G: both the ordinary and the strict p-cohomological dimension are monotone in a closed subgroup (NSW (3.3.5)).

For a discrete p-primary torsion U-module M, Shapiro's lemma identifies Hⁿ⁺¹(U, M) with Hⁿ⁺¹(G, Coind_U^G M) in every degree (TauCeti.ContinuousCohomology.bijective_shapiroMap), and the coinduced module is again a discrete p-primary torsion G-module (TauCeti.isPPrimaryTorsion_discreteCoind), so Hⁿ⁺¹(G, Coind_U^G M) vanishes when cd_p G ≤ n. Since the p-cohomological dimension of a compact group is detected in a single degree (TauCeti.cohomologicalDimensionLE_iff_forall_subsingleton_succ), this vanishing of Hⁿ⁺¹(U, -) is the statement cd_p U ≤ n. The coinduced module need not be finite when U is not open, which is why the test on all discrete p-primary torsion modules is the one used.

The case cd_p G ≤ 1 is what shows that closed subgroups of free pro-p groups are free pro-p.

The strict dimension uses Shapiro's lemma for all discrete coefficients rather than only the p-primary torsion ones. For an arbitrary discrete U-module M, the inverse of Shapiro's isomorphism Hⁱ(G, Coind_U^G M) ≅ Hⁱ(U, M) (TauCeti.ContinuousCohomology.shapiroIso) is additive and injective, so it carries the p-primary component of Hⁱ(U, M) injectively into that of Hⁱ(G, Coind_U^G M), which vanishes for i > n when scd_p G ≤ n. Here no single-degree reduction is needed, since the strict predicate already ranges over all discrete coefficients.

Main results

• TauCeti.CohomologicalDimensionLE.of_isClosed: the vanishing predicate cd_p U ≤ n for a closed subgroup U of a profinite group G with cd_p G ≤ n.
• TauCeti.cohomologicalDimensionAt_le_of_isClosed: cd_p U ≤ cd_p G in ℕ∞, for U closed in the profinite group G.
• TauCeti.StrictCohomologicalDimensionLE.of_isClosed: the strict vanishing predicate scd_p U ≤ n for a closed subgroup U of a profinite group G with scd_p G ≤ n.
• TauCeti.strictCohomologicalDimensionAt_le_of_isClosed: scd_p U ≤ scd_p G in ℕ∞, for U closed in the profinite group G; in particular for every open subgroup.

References

• J.-P. Serre, Galois Cohomology, Ch. I, §3.3, Prop. 14.
• J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields, 2nd ed., (3.3.5).
• L. Ribes, P. Zalesskii, Profinite Groups, 2nd ed., Thm. 7.3.1.

theorem TauCeti.CohomologicalDimensionLE.of_isClosed {p : ℕ} {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] {n : ℕ} (h : CohomologicalDimensionLE p G n) {U : Subgroup G} (hU : IsClosed ↑U) : CohomologicalDimensionLE p (↥U) n

cd_p ≤ n passes to closed subgroups, as the vanishing predicate: for U closed in a profinite group G with CohomologicalDimensionLE p G n, CohomologicalDimensionLE p U n.

theorem TauCeti.cohomologicalDimensionAt_le_of_isClosed {p : ℕ} {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] {U : Subgroup G} (hU : IsClosed ↑U) : cohomologicalDimensionAt p ↥U ≤ cohomologicalDimensionAt p G

cd_p U ≤ cd_p G for a closed subgroup U of a profinite group G: the p-cohomological dimension is monotone in a closed subgroup, as an inequality in ℕ∞.

theorem TauCeti.StrictCohomologicalDimensionLE.of_isClosed {p : ℕ} {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] {n : ℕ} (h : StrictCohomologicalDimensionLE p G n) {U : Subgroup G} (hU : IsClosed ↑U) : StrictCohomologicalDimensionLE p (↥U) n

scd_p ≤ n passes to closed subgroups, as the vanishing predicate: for U closed in a profinite group G with StrictCohomologicalDimensionLE p G n, StrictCohomologicalDimensionLE p U n.

theorem TauCeti.strictCohomologicalDimensionAt_le_of_isClosed {p : ℕ} {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] {U : Subgroup G} (hU : IsClosed ↑U) : strictCohomologicalDimensionAt p ↥U ≤ strictCohomologicalDimensionAt p G

scd_p U ≤ scd_p G for a closed subgroup U of a profinite group G (NSW (3.3.5)): the strict p-cohomological dimension is monotone in a closed subgroup, as an inequality in ℕ∞. In particular it applies to every open subgroup, which is closed.