The invariant part of H¹(N, 𝔽_p) and the rank of N ⧸ Nᵖ[N, G]

Let G be a profinite group, N a closed normal subgroup, and 𝔽_p the trivial G-module. The G-invariant classes in H¹(N, 𝔽_p) are the continuous homomorphisms N ⧸ Nᵖ[N, G] → 𝔽_p (TauCeti.ContCohomology.H1ConjInvariantsEquivOfSmulEqSelf), that is, the continuous 𝔽_p-dual of N ⧸ Nᵖ[N, G]. This quotient is a profinite group killed by p, hence pro-p whether or not G is, so by Burnside's basis theorem the dimension of its dual is its topological generator rank. So H¹(N, 𝔽_p)^G is finite exactly when N ⧸ Nᵖ[N, G] is topologically finitely generated, and then it has p ^ d(N ⧸ Nᵖ[N, G]) elements.

For a minimal presentation 1 → R → F → G → 1 of a pro-p group by a free pro-p group F, the transgression identifies H¹(R, 𝔽_p)^F with H²(G, 𝔽_p), and d(R ⧸ Rᵖ[R, F]) is the least number of generators of R as a closed normal subgroup of F. The count here is therefore what makes the dimension of H²(G, 𝔽_p) the relation rank of G.

Main results

• TauCeti.pLowerCentralStep_proPKernel: the relative elementary abelian p-quotient of the pro-p kernel is trivial, so H¹(R, M)^G vanishes for R the pro-p kernel and trivial coefficients M killed by p (TauCeti.subsingleton_h1ConjInvariants_proPKernel).
• TauCeti.finite_H1ConjInvariants_iff: H¹(N, 𝔽_p)^G is finite exactly when N ⧸ Nᵖ[N, G] is topologically finitely generated.
• TauCeti.natCard_H1ConjInvariants: in that case H¹(N, 𝔽_p)^G has p ^ d(N ⧸ Nᵖ[N, G]) elements.

References

• J. Neukirch, A. Schmidt and K. Wingberg, Cohomology of Number Fields, (3.9.1) and (3.9.5).
• J. P. Labute, Classification of Demushkin groups, Canad. J. Math. 19 (1967), §1.4.

The maximal pro-p kernel

theorem TauCeti.pLowerCentralStep_proPKernel {p : ℕ} {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] : pLowerCentralStep p (proPKernel p G) = proPKernel p G

The pro-p kernel has no nontrivial elementary abelian p-quotient invariant under the ambient group: Rᵖ[R,G] = R for R = proPKernel p G.

theorem TauCeti.subsingleton_h1ConjInvariants_proPKernel {p : ℕ} {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] {M : Type v} [AddCommGroup M] [TopologicalSpace M] [IsTopologicalAddGroup M] [T1Space M] [DistribMulAction G M] [ContinuousSMul G M] (htriv : ∀ (g : G) (m : M), g • m = m) (hpM : ∀ (m : M), p • m = 0) : Subsingleton ↥(ContCohomology.H1ConjInvariants G M (proPKernel p G))

The pro-p kernel has no invariant degree-one classes with trivial p-torsion coefficients. For R = proPKernel p G and a T1Space module M with trivial action and killed by p, H¹(R, M)^G is trivial.

Invariant degree-one classes

theorem TauCeti.finite_H1ConjInvariants_iff {p : ℕ} {G : Type u} [Group G] [TopologicalSpace G] [Fact (Nat.Prime p)] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] {N : Subgroup G} [N.Normal] [DistribMulAction G (ZMod p)] [ContinuousSMul G (ZMod p)] (hN : IsClosed ↑N) (htriv : ∀ (g : G) (m : ZMod p), g • m = m) : Finite ↥(ContCohomology.H1ConjInvariants G (ZMod p) N) ↔ IsTopologicallyFinitelyGenerated (↥N ⧸ (pLowerCentralStep p N).subgroupOf N)

Finiteness of H¹(N, 𝔽_p)^G. For a closed normal subgroup N of a profinite group G, the G-invariant part of H¹(N, 𝔽_p) is finite exactly when N ⧸ Nᵖ[N, G] is topologically finitely generated.

theorem TauCeti.natCard_H1ConjInvariants {p : ℕ} {G : Type u} [Group G] [TopologicalSpace G] [Fact (Nat.Prime p)] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] {N : Subgroup G} [N.Normal] [DistribMulAction G (ZMod p)] [ContinuousSMul G (ZMod p)] (hN : IsClosed ↑N) (htriv : ∀ (g : G) (m : ZMod p), g • m = m) (h : IsTopologicallyFinitelyGenerated (↥N ⧸ (pLowerCentralStep p N).subgroupOf N)) : Nat.card ↥(ContCohomology.H1ConjInvariants G (ZMod p) N) = p ^ topologicalGeneratorRankNat (↥N ⧸ (pLowerCentralStep p N).subgroupOf N) h

H¹(N, 𝔽_p)^G counts the generators of N ⧸ Nᵖ[N, G]. For a closed normal subgroup N of a profinite group G with N ⧸ Nᵖ[N, G] topologically finitely generated, the G-invariant part of H¹(N, 𝔽_p) has p ^ d(N ⧸ Nᵖ[N, G]) elements, where d is the topological generator rank.