The strict cohomological dimension exceeds the ordinary one by at most one

For a compact group G and a natural number p, the strict p-cohomological dimension is at most one more than the ordinary one, scd_p G ≤ cd_p G + 1 (NSW (3.3.3)). Together with TauCeti.cohomologicalDimensionAt_le_strictCohomologicalDimensionAt this pins scd_p G to one of the two values cd_p G and cd_p G + 1; both occur, ℤ_p having cd_p = 1 and scd_p = 2.

The argument is Serre's. Suppose Hⁱ(G, A) = 0 for every i > n and every discrete p-primary torsion G-module A, and let M be an arbitrary discrete G-module. Multiplication by p on M factors as M ↠ pM ↪ M, which gives two short exact sequences

0 → M[p] → M → pM → 0,        0 → pM → M → M ⧸ pM → 0,

whose outer terms M[p] and M ⧸ pM are killed by p, hence p-primary torsion. For i > n + 1 the long exact sequences make Hⁱ(G, M) → Hⁱ(G, pM) injective, because Hⁱ(G, M[p]) = 0, and Hⁱ(G, pM) → Hⁱ(G, M) injective, because Hⁱ⁻¹(G, M ⧸ pM) = 0. Their composite is multiplication by p on Hⁱ(G, M), which is therefore injective, so no nonzero class of Hⁱ(G, M) is killed by a power of p: the p-primary component of Hⁱ(G, M) vanishes.

The long exact sequence of continuous cohomology is available for coefficients in the universe of G, so the statements here fix the auxiliary universe parameter of the predicates of TauCeti.RepresentationTheory.Homological.ContCohomology.CohomologicalDimension.Basic to that of G, as in TauCeti.cohomologicalDimensionLE_iff_forall_subsingleton_succ.

Main results

• TauCeti.CohomologicalDimensionLE.nsmul_right_injective: under CohomologicalDimensionLE p G n, multiplication by p is injective on Hⁱ(G, M) for every i > n + 1 and every discrete M.
• TauCeti.CohomologicalDimensionLE.strictCohomologicalDimensionLE_add_one: CohomologicalDimensionLE p G n implies StrictCohomologicalDimensionLE p G (n + 1).
• TauCeti.strictCohomologicalDimensionAt_le_cohomologicalDimensionAt_add_one: scd_p G ≤ cd_p G + 1 in ℕ∞, including the case cd_p G = ⊤.
• TauCeti.StrictCohomologicalDimensionLE.primaryComponent_eq_bot_of_isSmoothDiscrete: the strict vanishing predicate read on a smooth discrete object of TopRep ℤ G, such as the trivial module ℤ.

References

• J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields, 2nd ed., (3.3.3).
• J.-P. Serre, Galois Cohomology, Ch. I, §3.2, Prop. 13.

theorem TauCeti.StrictCohomologicalDimensionLE.primaryComponent_eq_bot_of_isSmoothDiscrete {p : ℕ} {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] {n : ℕ} (h : StrictCohomologicalDimensionLE p G n) (X : TopRep ℤ G) (hX : IsSmoothDiscrete ℤ X) {i : ℕ} (hi : n < i) : AddCommGroup.primaryComponent (↑(continuousCohomology i X).toModuleCat) p = ⊥

The strict vanishing predicate on a smooth discrete coefficient object. Under StrictCohomologicalDimensionLE p G n, the p-primary component of Hⁱ(G, X) vanishes for every i > n and every smooth discrete object X of TopRep ℤ G, not only for the objects ofDiscreteModule ℤ G M the predicate is stated with.

theorem TauCeti.CohomologicalDimensionLE.nsmul_right_injective {p : ℕ} {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] {n : ℕ} (h : CohomologicalDimensionLE p G n) (M : Type u) [AddCommGroup M] [TopologicalSpace M] [DiscreteTopology M] [DistribMulAction G M] [ContinuousSMul G M] {i : ℕ} (hi : n + 1 < i) : Function.Injective fun (x : ↑(continuousCohomology i (ofDiscreteModule ℤ G M)).toModuleCat) => p • x

Multiplication by p is injective above cd_p + 1. For a compact group G with CohomologicalDimensionLE p G n, multiplication by p is injective on Hⁱ(G, M) for every i > n + 1 and every discrete G-module M with continuous action, p-primary or not.

theorem TauCeti.CohomologicalDimensionLE.strictCohomologicalDimensionLE_add_one {p : ℕ} {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] {n : ℕ} (h : CohomologicalDimensionLE p G n) : StrictCohomologicalDimensionLE p G (n + 1)

cd_p ≤ n implies scd_p ≤ n + 1 (NSW (3.3.3)). For a compact group G, if Hⁱ(G, A) vanishes for every i > n and every discrete p-primary torsion G-module A, then the p-primary component of Hⁱ(G, M) vanishes for every i > n + 1 and every discrete G-module M.

theorem TauCeti.strictCohomologicalDimensionAt_le_cohomologicalDimensionAt_add_one (p : ℕ) (G : Type u) [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] : strictCohomologicalDimensionAt p G ≤ cohomologicalDimensionAt p G + 1

scd_p ≤ cd_p + 1 (NSW (3.3.3)): over a compact group, the strict p-cohomological dimension exceeds the p-cohomological dimension by at most one. The case cd_p G = ⊤ is included, the right-hand side then being ⊤ + 1 = ⊤.