Inflation with twisted coefficients and the prescription property

Let N be a normal subgroup of a topological group G, let π : G → G ⧸ N be the quotient map and let χ : G ⧸ N →ₜ* ℤ_pˣ be a continuous character. The subgroup N acts trivially on the twisted module I(χ ∘ π)/pⁱ, so its N-invariants are all of it, and they are I(χ)/pⁱ as a G ⧸ N-module. Reading the degree-one inflation H¹(G ⧸ N, M ^ N) → H¹(G, M) through this identification gives the twisted inflation

H¹(G ⧸ N, I(χ)/pⁱ) → H¹(G, I(χ ∘ π)/pⁱ),

which is injective and commutes with the reductions I(χ)/pⁱ → I(χ)/pʲ.

If moreover N has no nontrivial continuous p-group quotient, the twisted inflation is bijective: by the inflation-restriction sequence its image is the kernel of restriction to N, and H¹(N, I(χ ∘ π)/pⁱ) vanishes, because N acts trivially, so that its classes are the continuous homomorphisms from N to the finite p-group ℤ/pⁱ, and there are none but 0. Consequently χ has the prescription property exactly when χ ∘ π has it. This applies to the maximal pro-p quotient G(p) = G ⧸ proPKernel p G of a profinite group G, whose pro-p kernel has no p-quotient (TauCeti.proPKernel_proPKernel_eq_top): a character of G(p) has the prescription property if and only if its pullback to G does. For G an absolute Galois group, this reduces the prescription property of a character of the maximal pro-p Galois group, such as a descended cyclotomic character, to the corresponding statement about the absolute Galois group itself, where Kummer theory is available.

Main definitions

• TauCeti.ZModTwist.fixedPointsEquiv: the identification of the N-invariants of I(χ ∘ π)/pⁱ with I(χ)/pⁱ, equivariant for G ⧸ N.
• TauCeti.explicitInfl1ZModTwist: twisted inflation H¹(G ⧸ N, I(χ)/pⁱ) → H¹(G, I(χ ∘ π)/pⁱ).

Main results

• TauCeti.explicitInfl1ZModTwist_eq_explicitMap1: twisted inflation is the pullback along G → G ⧸ N, with the identity on residue classes as coefficient map.
• TauCeti.explicitInfl1ZModTwist_injective: twisted inflation is injective.
• TauCeti.explicitCoeff1_reduce_comp_explicitInfl1ZModTwist: twisted inflation commutes with the reductions between levels.
• TauCeti.explicitInfl1ZModTwist_bijective: twisted inflation is bijective when N has no nontrivial continuous p-group quotient.
• TauCeti.hasPrescriptionProperty_comp_quotientMk_iff: χ ∘ π has the prescription property exactly when χ does, under the same hypothesis on N.
• TauCeti.hasPrescriptionProperty_comp_quotientMk_proPKernel_iff: the case of the maximal pro-p quotient of a profinite group.

References

• J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields, 2nd ed., (1.6.7).
• J. P. Labute, Classification of Demushkin groups, Canad. J. Math. 19 (1967), 106–132, §2.

theorem TauCeti.ZModTwist.smul_eq_self_of_mem {p : ℕ} [Fact (Nat.Prime p)] {G : Type u} [Group G] [TopologicalSpace G] (N : Subgroup G) [N.Normal] (χ : G ⧸ N →ₜ* ℤ_[p]ˣ) (i : ℕ) {n : G} (hn : n ∈ N) (x : ZModTwist (χ.comp (ContinuousMonoidHom.quotientMk N)) i) : n • x = x

The normal subgroup N acts trivially on the twisted module of a character pulled back from G ⧸ N, because the pulled-back character is 1 on N.

def TauCeti.ZModTwist.fixedPointsEquiv {p : ℕ} [Fact (Nat.Prime p)] {G : Type u} [Group G] [TopologicalSpace G] (N : Subgroup G) [N.Normal] (χ : G ⧸ N →ₜ* ℤ_[p]ˣ) (i : ℕ) : ↥(FixedPoints.addSubgroup (↥N) (ZModTwist (χ.comp (ContinuousMonoidHom.quotientMk N)) i)) ≃+ ZModTwist χ i

The N-invariants of I(χ ∘ π)/pⁱ, which are all of it (smul_eq_self_of_mem), are I(χ)/pⁱ. On residue classes this is the identity; it is G ⧸ N-equivariant (fixedPointsEquiv_smul).

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem TauCeti.ZModTwist.val_fixedPointsEquiv {p : ℕ} [Fact (Nat.Prime p)] {G : Type u} [Group G] [TopologicalSpace G] (N : Subgroup G) [N.Normal] (χ : G ⧸ N →ₜ* ℤ_[p]ˣ) (i : ℕ) (x : ↥(FixedPoints.addSubgroup (↥N) (ZModTwist (χ.comp (ContinuousMonoidHom.quotientMk N)) i))) : ((fixedPointsEquiv N χ i) x).val = (↑x).val

The identification of the invariants is the identity on residue classes.

@[simp]

theorem TauCeti.ZModTwist.val_coe_fixedPointsEquiv_symm {p : ℕ} [Fact (Nat.Prime p)] {G : Type u} [Group G] [TopologicalSpace G] (N : Subgroup G) [N.Normal] (χ : G ⧸ N →ₜ* ℤ_[p]ˣ) (i : ℕ) (y : ZModTwist χ i) : (↑((fixedPointsEquiv N χ i).symm y)).val = y.val

The inverse identification of the invariants is the identity on residue classes.

theorem TauCeti.ZModTwist.fixedPointsEquiv_smul {p : ℕ} [Fact (Nat.Prime p)] {G : Type u} [Group G] [TopologicalSpace G] (N : Subgroup G) [N.Normal] (χ : G ⧸ N →ₜ* ℤ_[p]ˣ) (i : ℕ) (q : G ⧸ N) (x : ↥(FixedPoints.addSubgroup (↥N) (ZModTwist (χ.comp (ContinuousMonoidHom.quotientMk N)) i))) : (fixedPointsEquiv N χ i) (q • x) = q • (fixedPointsEquiv N χ i) x

The identification of the invariants is equivariant for the quotient action of G ⧸ N.

noncomputable def TauCeti.explicitInfl1ZModTwist {p : ℕ} [Fact (Nat.Prime p)] {G : Type u} [Group G] [TopologicalSpace G] (N : Subgroup G) [N.Normal] (χ : G ⧸ N →ₜ* ℤ_[p]ˣ) (i : ℕ) : ContCohomology.H1 (G ⧸ N) (ZModTwist χ i) →+ ContCohomology.H1 G (ZModTwist (χ.comp (ContinuousMonoidHom.quotientMk N)) i)

Twisted inflation H¹(G ⧸ N, I(χ)/pⁱ) → H¹(G, I(χ ∘ π)/pⁱ): degree-one inflation TauCeti.ContCohomology.explicitInfl1, read through the identification TauCeti.ZModTwist.fixedPointsEquiv of the N-invariants of I(χ ∘ π)/pⁱ with I(χ)/pⁱ.

Equations

• One or more equations did not get rendered due to their size.

theorem TauCeti.explicitInfl1ZModTwist_eq_explicitMap1 {p : ℕ} [Fact (Nat.Prime p)] {G : Type u} [Group G] [TopologicalSpace G] (N : Subgroup G) [N.Normal] (χ : G ⧸ N →ₜ* ℤ_[p]ˣ) (i : ℕ) : explicitInfl1ZModTwist N χ i = ContCohomology.explicitMap1 (G ⧸ N) (ZModTwist χ i) G (ZModTwist (χ.comp (ContinuousMonoidHom.quotientMk N)) i) (ContinuousMonoidHom.quotientMk N) (ZModTwist.compEquiv (ContinuousMonoidHom.quotientMk N) χ i).toAddMonoidHom ⋯ ⋯

Twisted inflation is the pullback along the quotient map G → G ⧸ N, paired with the identification TauCeti.ZModTwist.compEquiv of I(χ)/pⁱ with I(χ ∘ π)/pⁱ, which is the identity on residue classes.

theorem TauCeti.explicitInfl1ZModTwist_injective {p : ℕ} [Fact (Nat.Prime p)] {G : Type u} [Group G] [TopologicalSpace G] (N : Subgroup G) [N.Normal] (χ : G ⧸ N →ₜ* ℤ_[p]ˣ) (i : ℕ) : Function.Injective ⇑(explicitInfl1ZModTwist N χ i)

Twisted inflation is injective.

theorem TauCeti.explicitCoeff1_reduce_comp_explicitInfl1ZModTwist {p : ℕ} [Fact (Nat.Prime p)] {G : Type u} [Group G] [TopologicalSpace G] (N : Subgroup G) [N.Normal] (χ : G ⧸ N →ₜ* ℤ_[p]ˣ) (i : ℕ) {j : ℕ} (hij : j ≤ i) : (ContCohomology.explicitCoeff1 G (ZModTwist (χ.comp (ContinuousMonoidHom.quotientMk N)) i) (ZModTwist.reduce (χ.comp (ContinuousMonoidHom.quotientMk N)) hij) ⋯).comp (explicitInfl1ZModTwist N χ i) = (explicitInfl1ZModTwist N χ j).comp (ContCohomology.explicitCoeff1 (G ⧸ N) (ZModTwist χ i) (ZModTwist.reduce χ hij) ⋯)

Twisted inflation commutes with the reductions I(χ)/pⁱ → I(χ)/pʲ: on cocycles both composites send a cocycle on G ⧸ N to the same cocycle on G.

theorem TauCeti.explicitInfl1ZModTwist_bijective {p : ℕ} [Fact (Nat.Prime p)] {G : Type u} [Group G] [TopologicalSpace G] (N : Subgroup G) [N.Normal] (χ : G ⧸ N →ₜ* ℤ_[p]ˣ) (i : ℕ) (hN : proPKernel p ↥N = ⊤) : Function.Bijective ⇑(explicitInfl1ZModTwist N χ i)

Twisted inflation is bijective when N has no nontrivial continuous p-group quotient. Its image is the kernel of restriction to N (TauCeti.ContCohomology.explicitInfRes_exact), and H¹(N, I(χ ∘ π)/pⁱ) vanishes: N acts trivially, so its classes are continuous homomorphisms from N to the finite p-group ℤ/pⁱ (TauCeti.ContCohomology.H1EquivOfSmulEqSelf), and these are trivial (TauCeti.eq_one_of_proPKernel_eq_top).

theorem TauCeti.hasPrescriptionProperty_comp_quotientMk_iff {p : ℕ} [Fact (Nat.Prime p)] {G : Type u} [Group G] [TopologicalSpace G] (N : Subgroup G) [N.Normal] (χ : G ⧸ N →ₜ* ℤ_[p]ˣ) (hN : proPKernel p ↥N = ⊤) : HasPrescriptionProperty (χ.comp (ContinuousMonoidHom.quotientMk N)) ↔ HasPrescriptionProperty χ

The prescription property descends along a quotient with no p-quotient kernel. If N has no nontrivial continuous p-group quotient, a character χ of G ⧸ N has the prescription property exactly when its pullback χ ∘ π to G does: twisted inflation identifies the reductions H¹(-, I(-)/pⁱ) → H¹(-, I(-)/p) of the two groups.

theorem TauCeti.hasPrescriptionProperty_comp_quotientMk_proPKernel_iff {p : ℕ} [Fact (Nat.Prime p)] {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] (χ : maximalProPQuotient p G →ₜ* ℤ_[p]ˣ) : HasPrescriptionProperty (χ.comp (ContinuousMonoidHom.quotientMk (proPKernel p G))) ↔ HasPrescriptionProperty χ

The prescription property on the maximal pro-p quotient. For a profinite group G, a character of its maximal pro-p quotient G(p) has the prescription property exactly when its pullback to G does.