Transgression for extensions inside the Frattini subgroup

Let G be a profinite group, p a prime, and N a closed normal subgroup contained in the pro-p Frattini subgroup Φ(G) = proPFrattini p G. Let M be a discrete abelian group killed by p on which G acts trivially, for instance 𝔽_p. Then every continuous 1-cocycle on G with values in M is a continuous homomorphism to an elementary abelian p-group, so it vanishes on Φ(G) and hence on N: restriction H¹(G, M) → H¹(N, M) is zero. By exactness of the five-term sequence

0 → H¹(G ⧸ N, M ^ N) → H¹(G, M) → H¹(N, M) ^ (G ⧸ N) → H²(G ⧸ N, M ^ N) → H²(G, M),

the transgression H¹(N, M) ^ (G ⧸ N) → H²(G ⧸ N, M ^ N) is then injective, and it is bijective as soon as H²(G, M) vanishes.

The free pro-p specialization is in TauCeti.Topology.Algebra.Group.Profinite.Free.Transgression.

Main results

• TauCeti.explicitRes1_eq_zero_of_le_proPFrattini: restriction on H¹ with trivial coefficients killed by p vanishes on a subgroup of the pro-p Frattini subgroup.
• TauCeti.transgression_injective_of_le_proPFrattini: the transgression of such a subgroup is injective.
• TauCeti.transgression_bijective_of_le_proPFrattini: it is bijective when H²(G, M) = 0.

References

• J. Neukirch, A. Schmidt and K. Wingberg, Cohomology of Number Fields, 2nd ed., (3.9.5).
• J. P. Labute, Classification of Demushkin groups, Canad. J. Math. 19 (1967), §1.4.
• J.-P. Serre, Galois Cohomology, Chapter I, §4.3.

theorem TauCeti.explicitRes1_eq_zero_of_le_proPFrattini {p : ℕ} [Fact (Nat.Prime p)] {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] {M : Type v} [AddCommGroup M] [TopologicalSpace M] [IsTopologicalAddGroup M] [DiscreteTopology M] [DistribMulAction G M] [ContinuousSMul G M] {N : Subgroup G} (hN : N ≤ proPFrattini p G) (htriv : ∀ (g : G) (m : M), g • m = m) (hpM : ∀ (m : M), p • m = 0) : ContCohomology.explicitRes1 G M N = 0

Restriction to a subgroup of the Frattini subgroup vanishes. For a profinite group G, a subgroup N ≤ proPFrattini p G, and a discrete abelian group M killed by p with trivial action, restriction H¹(G, M) → H¹(N, M) is zero.

theorem TauCeti.transgression_injective_of_le_proPFrattini {p : ℕ} [Fact (Nat.Prime p)] {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] {M : Type v} [AddCommGroup M] [TopologicalSpace M] [IsTopologicalAddGroup M] [DiscreteTopology M] [DistribMulAction G M] [ContinuousSMul G M] {N : Subgroup G} [N.Normal] (hNc : IsClosed ↑N) (hN : N ≤ proPFrattini p G) (htriv : ∀ (g : G) (m : M), g • m = m) (hpM : ∀ (m : M), p • m = 0) : Function.Injective ⇑(ContCohomology.transgression G M N hNc)

Transgression is injective below the Frattini subgroup. For a closed normal subgroup N ≤ proPFrattini p G of a profinite group and a discrete abelian group M killed by p with trivial action, the transgression H¹(N, M) ^ (G ⧸ N) → H²(G ⧸ N, M ^ N) is injective.

theorem TauCeti.transgression_bijective_of_le_proPFrattini {p : ℕ} [Fact (Nat.Prime p)] {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] {M : Type v} [AddCommGroup M] [TopologicalSpace M] [IsTopologicalAddGroup M] [DiscreteTopology M] [DistribMulAction G M] [ContinuousSMul G M] {N : Subgroup G} [N.Normal] [Subsingleton (ContCohomology.H2 G M)] (hNc : IsClosed ↑N) (hN : N ≤ proPFrattini p G) (htriv : ∀ (g : G) (m : M), g • m = m) (hpM : ∀ (m : M), p • m = 0) : Function.Bijective ⇑(ContCohomology.transgression G M N hNc)

Transgression is bijective below the Frattini subgroup when H²(G, M) vanishes. For a closed normal subgroup N ≤ proPFrattini p G of a profinite group and a discrete abelian group M killed by p with trivial action and H²(G, M) = 0, the transgression H¹(N, M) ^ (G ⧸ N) → H²(G ⧸ N, M ^ N) is bijective.