H⁰ is co-effaceable on the finite modules of an infinite pro-p group

Let G be an infinite pro-p group and M a finite discrete p-primary torsion G-module. Then M is a quotient of a finite discrete G-module M₁ in such a way that no nonzero invariant of M lifts to an invariant of M₁: the map H⁰(G, M₁) → H⁰(G, M) induced by the surjection M₁ → M is zero. The functor H⁰ is co-effaceable on this category of coefficients.

The module M₁ is the coinduced module Coind_V^G M of TauCeti.DiscreteCoind along an open subgroup V, and the surjection is the trace TauCeti.DiscreteCoind.trace. The choice of V is the whole content. Fix an open normal subgroup U acting trivially on M. Since G is infinite and pro-p, U contains open normal subgroups V of arbitrarily large p-power relative index [U : V] (TauCeti.IsProP.exists_openNormalSubgroup_le_pow_dvd_relIndex). A G-invariant element of Coind_V^G M is a constant function with value m ∈ M, and its trace is the norm ∑_{q ∈ G ⧸ V} q.out • m, which groups into [U : V] copies of the norm along G ⧸ U because U fixes m (TauCeti.DiscreteCoind.trace_eq_relIndex_nsmul_of_forall_smul_eq). Once [U : V] is divisible by the exponent of M, this vanishes.

Co-effaceability of H⁰ is the step of Tate's duality argument for the cohomological dimension of a Demushkin group that upgrades the surjectivity of the duality map H⁰(G, M) → H²(G, M^∨)^∨ to injectivity: the connecting map of 0 → K → Coind_V^G M → M → 0 embeds H⁰(G, M) into H¹(G, K), where the duality map is already known to be bijective. The hypothesis that G is infinite is used and cannot be dropped: for a finite group G the module 𝔽_p[G] is projective, so every G-equivariant surjection onto it splits and its invariants lift. In the proof, infinitude is what supplies open normal subgroups of arbitrarily large p-power relative index.

Main results

• TauCeti.IsProP.exists_isOpen_trace_eq_zero_of_mem_H0: co-effaceability of H⁰. For a finite discrete p-primary torsion module M over an infinite pro-p group there is an open subgroup V of finite index such that the trace Coind_V^G M → M vanishes on the G-invariants.

References

• J.-P. Serre, Structure de certains pro-p-groupes (d'après Demuškin), Séminaire Bourbaki 8 (1962–1964), exp. 252, §9.1.
• J.-P. Serre, Galois Cohomology, Ch. I, §4.5.

theorem TauCeti.IsProP.exists_isOpen_trace_eq_zero_of_mem_H0 {p : ℕ} [Fact (Nat.Prime p)] {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] [Infinite G] (hG : IsProP p G) (M : Type v) [AddCommGroup M] [TopologicalSpace M] [DiscreteTopology M] [DistribMulAction G M] [ContinuousSMul G M] [Finite M] (hM : IsAddPGroup p M) : ∃ (V : Subgroup G) (x : V.FiniteIndex), IsOpen ↑V ∧ ∀ f ∈ ContCohomology.H0 G (DiscreteCoind G V M), (DiscreteCoind.trace G V M) f = 0

H⁰ is co-effaceable on the finite modules of an infinite pro-p group. For a finite discrete p-primary torsion G-module M there is an open subgroup V of finite index such that the trace Coind_V^G M → M, a G-equivariant surjection, vanishes on the G-invariants: the induced map H⁰(G, Coind_V^G M) → H⁰(G, M) is zero.