Vanishing of H¹ of the pro-p class module for p-group quotients

Let G be a profinite group with scd_p G ≤ 2 and let V be an open normal subgroup such that G ⧸ V is a finite p-group. Then H¹(G ⧸ V, V^ab(p)) = 0.

This is the degree-one half of the class-module axioms for V^ab(p) in the p-group case of NSW (3.6.4), (ii) ⇒ (iii), extending the prime-order case by the reduction of NSW (3.6.3) through a central subgroup of order p. It is the input for the corresponding statement about the class u_{G/V}(p) in H² for p-group quotients, and then for the Sylow reduction to arbitrary finite quotients G ⧸ V.

Main result

• TauCeti.subsingleton_h1_abelianizationProP_of_isPGroup: H¹(G ⧸ V, V^ab(p)) = 0 when G ⧸ V is a finite p-group.

References

• J. Neukirch, A. Schmidt and K. Wingberg, Cohomology of Number Fields, 2nd ed., (3.6.3) and the proof of (3.6.4), (ii) ⇒ (iii).

theorem TauCeti.subsingleton_h1_abelianizationProP_of_le {p : ℕ} {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] {V W : Subgroup G} [V.Normal] [W.Normal] (hp : Nat.Prime p) (h : strictCohomologicalDimensionAt p G ≤ 2) (hVW : V ≤ W) (hV : IsOpen ↑V) [Subsingleton (ContCohomology.H1 (G ⧸ W) (Additive (abelianizationProP p G W)))] [Subsingleton (ContCohomology.H1 (↥W ⧸ V.subgroupOf W) (Additive (abelianizationProP p (↥W) (V.subgroupOf W))))] : Subsingleton (ContCohomology.H1 (G ⧸ V) (Additive (abelianizationProP p G V)))

For open normal subgroups V ≤ W of G with scd_p G ≤ 2, if H¹(G ⧸ W, W^ab(p)) and H¹(W ⧸ V, V^ab(p)) vanish, then so does H¹(G ⧸ V, V^ab(p)).

theorem TauCeti.subsingleton_h1_abelianizationProP_of_isPGroup {p : ℕ} {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] {V : Subgroup G} [V.Normal] (hp : Nat.Prime p) (hG : strictCohomologicalDimensionAt p G ≤ 2) (hV : IsOpen ↑V) (hpV : IsPGroup p (G ⧸ V)) : Subsingleton (ContCohomology.H1 (G ⧸ V) (Additive (abelianizationProP p G V)))

The class module of a finite p-group quotient has trivial H¹. For a profinite group G with scd_p G ≤ 2 and an open normal subgroup V whose quotient is a p-group, H¹(G ⧸ V, V^ab(p)) = 0. This is the degree-one part of NSW (3.6.4), (ii) ⇒ (iii), in the p-group case reduced to prime order as in NSW (3.6.3).