Vanishing of H¹ of the pro-p class module for a cyclic quotient

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 cyclic p-group. Then H¹(G ⧸ V, V^ab(p)) = 0. This is the degree-one half of NSW (3.6.4), (ii) ⇒ (iii), for a cyclic quotient. The vanishing of H¹ is the analogue of Hilbert's Theorem 90 among the class formation axioms, so it is one of the two conditions that make the modules V^ab(p) a p-class formation for G. The prime-order case is the base case of the induction over p-group quotients in NSW (3.6.4).

Main results

• TauCeti.subsingleton_h1_abelianizationProP_of_isCyclic: H¹(G ⧸ V, V^ab(p)) = 0 for a cyclic p-group quotient G ⧸ V.
• TauCeti.subsingleton_h1_abelianizationProP_of_card_eq_prime: the case #(G ⧸ V) = p, the base case of the induction over p-group quotients.

References

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

theorem TauCeti.subsingleton_h1_abelianizationProP_of_isCyclic {p : ℕ} {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] {V : Subgroup G} [V.Normal] (hp : Nat.Prime p) (h : strictCohomologicalDimensionAt p G ≤ 2) (hV : IsOpen ↑V) (hpV : IsPGroup p (G ⧸ V)) [IsCyclic (G ⧸ V)] : Subsingleton (ContCohomology.H1 (G ⧸ V) (Additive (abelianizationProP p G V)))

The class module of a cyclic quotient has trivial H¹ (NSW (3.6.4), (ii) ⇒ (iii), in degree one and p-primary form). For a profinite group G with scd_p G ≤ 2 and an open normal subgroup V whose quotient G ⧸ V is a cyclic p-group, H¹(G ⧸ V, V^ab(p)) = 0.

theorem TauCeti.subsingleton_h1_abelianizationProP_of_card_eq_prime {p : ℕ} {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] {V : Subgroup G} [V.Normal] (hp : Nat.Prime p) (h : strictCohomologicalDimensionAt p G ≤ 2) (hV : IsOpen ↑V) (hcard : Nat.card (G ⧸ V) = p) : Subsingleton (ContCohomology.H1 (G ⧸ V) (Additive (abelianizationProP p G V)))

The class module of a quotient of prime order has trivial H¹. For a profinite group G with scd_p G ≤ 2 and an open normal subgroup V of prime index p, H¹(G ⧸ V, V^ab(p)) = 0. This is the base case of the induction over p-group quotients in NSW (3.6.4).