The transfer to the pro-p class module in strict cohomological dimension two

Let G be a profinite group with scd_p G ≤ 2 and let V be an open subgroup. The transfer Ver : G → V^ab(p) (TauCeti.abelianizationProPTransfer) has the same kernel as the canonical map G → G^ab(p), so it induces an injection G^ab(p) → V^ab(p); for V normal, its image is the subgroup of G ⧸ V-invariants. Together these are NSW (3.6.4)(ii) in p-primary form, G^ab(p) ≅ V^ab(p)^{G/V}.

One inclusion of the kernels needs no hypothesis on G beyond profiniteness: Ver is a continuous homomorphism to the abelian pro-p group V^ab(p), so it factors through G^ab(p) (TauCeti.abelianizationProPTransfer_eq_one_of_mk_eq_one). The other inclusion is NSW (3.3.11) in degree 2 with coefficients ℤ, read on characters. A continuous character χ : G → ℤ/p^k has a p-primary integral Bockstein δχ ∈ H²(G, ℤ). Because scd_p G ≤ 2, δχ is a corestriction from H²(V, ℤ), so it is the corestriction of the Bockstein δψ of a character ψ of V. Corestriction commutes with the Bockstein, and on characters it is the transfer (NSW (1.5.9)), so δχ = δ(ψ ∘ Ver). Moving both characters to a common modulus, and using that the degree-one Bockstein is injective, gives χ = ψ ∘ Ver up to a multiplication of the coefficients. So χ kills the kernel of Ver (TauCeti.zmodChar_eq_one_of_transfer_eq_one). Finally, continuous characters into the groups ℤ/p^k separate the points of the abelian pro-p group G^ab(p) (TauCeti.exists_continuous_zmodChar_of_notMem). For the kernel, V need not be normal.

The image is the other half of (3.3.11), injectivity of corestriction on the coinvariants. The image of Ver is invariant (TauCeti.smul_abelianizationProPTransfer). Conversely, let ψ : V^ab(p) → ℤ/p^k be a continuous character that is trivial on the image of Ver. Read on V, its class in H¹(V, ℤ/p^k) has trivial corestriction, since that corestriction is ψ ∘ Ver, so its integral Bockstein lies in the subgroup generated by the differences (g)_* w - w of p-primary classes w ∈ H²(V, ℤ). Each such w is the Bockstein of a character of p-power order of V. That character factors through V^ab(p), so (g)_* w - w restricts to zero on the closed subgroup Z of the elements of V whose classes are invariant. Hence the Bockstein of ψ restricted to Z vanishes, and so does that restriction itself, the Bockstein being injective in degree one. This shows that ψ kills the invariants (TauCeti.zmodChar_eq_one_of_comp_transfer_eq_one). The image of Ver is compact, hence closed, and characters separate it from any invariant element outside it.

Main results

• TauCeti.zmodChar_eq_one_of_transfer_eq_one: under scd_p G ≤ 2, a continuous character of G of p-power order kills the kernel of the transfer.
• TauCeti.abelianizationProPTransfer_eq_one_iff: under scd_p G ≤ 2, the kernel of the transfer is the kernel of G → G^ab(p).
• TauCeti.zmodChar_eq_one_of_comp_transfer_eq_one: under scd_p G ≤ 2 and for V normal, a continuous character of V^ab(p) of p-power order that kills the image of the transfer kills the G ⧸ V-invariants.
• TauCeti.abelianizationProPTransfer_range: under scd_p G ≤ 2 and for V normal, the image of the transfer is the subgroup of G ⧸ V-invariants of V^ab(p).

References

• J. Neukirch, A. Schmidt and K. Wingberg, Cohomology of Number Fields, 2nd ed., (1.5.9), (3.3.11) and the proof of (3.6.4), (i) ⇒ (ii).

theorem TauCeti.zmodChar_eq_one_of_transfer_eq_one {p : ℕ} {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] {V : Subgroup G} [V.FiniteIndex] (hp : Nat.Prime p) (h : strictCohomologicalDimensionAt p G ≤ 2) (hV : IsOpen ↑V) {k : ℕ} (χ : G →* Multiplicative (ZMod (p ^ k))) (hχ : Continuous ⇑χ) {g : G} (hg : (abelianizationProPTransfer p G V) g = 1) : χ g = 1

A character of p-power order dies on the kernel of the transfer. For a profinite group G with scd_p G ≤ 2 and an open subgroup V, every continuous character χ : G → ℤ/p^k is trivial on the kernel of the transfer Ver : G → V^ab(p). This is NSW (3.3.11) at n = 2 with integral coefficients, read on characters through the integral Bockstein maps and (1.5.9).

theorem TauCeti.abelianizationProPTransfer_eq_one_iff {p : ℕ} {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] {V : Subgroup G} [V.FiniteIndex] (hp : Nat.Prime p) (h : strictCohomologicalDimensionAt p G ≤ 2) (hV : IsOpen ↑V) (g : G) : (abelianizationProPTransfer p G V) g = 1 ↔ (maximalProPQuotient.mk p (TopologicalAbelianization G)) ↑g = 1

The kernel of the transfer (NSW (3.6.4)(ii), injectivity, in p-primary form). For a profinite group G with scd_p G ≤ 2 and an open subgroup V, an element of G dies under the transfer Ver : G → V^ab(p) exactly when it dies in G^ab(p). So Ver induces an injection G^ab(p) → V^ab(p). The prime p is needed only for the forward direction.

theorem TauCeti.zmodChar_eq_one_of_comp_transfer_eq_one {p : ℕ} {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] {V : Subgroup G} [V.FiniteIndex] [V.Normal] (hp : Nat.Prime p) (h : strictCohomologicalDimensionAt p G ≤ 2) (hV : IsOpen ↑V) {k : ℕ} (ψ : abelianizationProP p G V →* Multiplicative (ZMod (p ^ k))) (hψ : Continuous ⇑ψ) (hψV : ∀ (g : G), ψ ((abelianizationProPTransfer p G V) g) = 1) {x : abelianizationProP p G V} (hx : ∀ (q : G ⧸ V), q • x = x) : ψ x = 1

A character that dies on the image of the transfer dies on the invariants. For a profinite group G with scd_p G ≤ 2 and an open normal subgroup V, a continuous character ψ : V^ab(p) → ℤ/p^k that is trivial on the image of the transfer Ver : G → V^ab(p) is trivial on every G ⧸ V-invariant element of V^ab(p). This is NSW (3.3.11) at n = 2 with integral coefficients, the injectivity of corestriction on the coinvariants, read on characters.

theorem TauCeti.abelianizationProPTransfer_range {p : ℕ} {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] {V : Subgroup G} [V.FiniteIndex] [V.Normal] (hp : Nat.Prime p) (h : strictCohomologicalDimensionAt p G ≤ 2) (hV : IsOpen ↑V) : (abelianizationProPTransfer p G V).range = FixedPoints.subgroup (G ⧸ V) (abelianizationProP p G V)

The image of the transfer (NSW (3.6.4)(ii), surjectivity, in p-primary form). For a profinite group G with scd_p G ≤ 2 and an open normal subgroup V, the image of the transfer Ver : G → V^ab(p) is the subgroup of G ⧸ V-invariants of V^ab(p). Together with TauCeti.abelianizationProPTransfer_eq_one_iff, Ver induces an isomorphism of G^ab(p) onto (V^ab(p))^{G/V}.