Surjections from relation modules over ℤ_p[G]

Let G be a finite group, Λ = ℤ_p[G] and I_G ⊆ Λ the augmentation ideal. Let 0 → A → Y → I_G → 0 be an exact sequence of Λ-modules, and φ : Λ^ι → Y a surjection from a free module of finite rank. For every generating family g : ι → G of the same size, the relation module relationModule ℤ_[p] G g, the kernel of Λ^ι → I_G, e_i ↦ g_i - 1, maps onto A with kernel isomorphic to the kernel of φ.

This is how an integral description of Y becomes a description of A by relation modules: the two surjections Λ^ι → I_G, through φ and through g, have isomorphic kernels by Schanuel's lemma and Krull–Schmidt cancellation over ℤ_p[G] (TauCeti.nonempty_ker_linearEquiv_of_range_eq), and φ carries the first kernel onto A. For A the p-adic completion of Lˣ of a finite Galois layer L/K of p-adic fields and Y the Tate module of the layer, this gives the surjection R^ab(p) ↠ A(L) with kernel ℤ_p[G] in the proof of NSW (7.4.1).

Conversely, a surjection β from the relation module of a generating family of size n + 2 onto a module A with A ⊗ ℚ_p ≃ ℚ_p[G]^n ⊕ ℚ_p has a kernel whose rationalization is ℚ_p[G]: the rationalized relation module is ℚ_p[G]^(n + 1) ⊕ ℚ_p (TauCeti.MonoidAlgebra.nonempty_relationModule_tensor_linearEquiv), and rational representations cancel. If that kernel is projective, Swan's theorem (TauCeti.nonempty_linearEquiv_of_projective_of_tensorRat) identifies it with ℤ_p[G]. For n = [K : ℚ_p] and A the p-adic completion of Lˣ, whose rationalization is ℚ_p[G]^n ⊕ ℚ_p (TauCeti.nonempty_padicCompletionUnits_tensorRat_linearEquiv), this is the step of NSW (7.4.2)(i) that identifies the kernel of R^ab(p) ↠ A(L) with ℤ_p[G] once it is known to be projective.

Main results

• TauCeti.exists_relationModule_surjective_of_exact: the relation module of a generating family maps onto the kernel A of an extension 0 → A → Y → I_G → 0, with kernel isomorphic to that of a given surjection Λ^ι → Y.
• TauCeti.nonempty_ker_linearEquiv_of_relationModule_surjective: a surjection from the relation module of n + 2 generators onto a module with rationalization ℚ_p[G]^n ⊕ ℚ_p and projective kernel has kernel isomorphic to ℤ_p[G].

References

• J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields, second edition, (5.6.6), (5.6.10)(ii), and the proofs of (7.4.1) and (7.4.2).

theorem TauCeti.exists_relationModule_surjective_of_exact {p : ℕ} [Fact (Nat.Prime p)] {G : Type u_1} [Group G] [Finite G] {ι : Type u_2} [Fintype ι] {A : Type u_3} {Y : Type u_4} [AddCommGroup A] [Module (MonoidAlgebra ℤ_[p] G) A] [AddCommGroup Y] [Module (MonoidAlgebra ℤ_[p] G) Y] {i : A →ₗ[MonoidAlgebra ℤ_[p] G] Y} {π : Y →ₗ[MonoidAlgebra ℤ_[p] G] ↥(RingHom.ker (MonoidAlgebra.augmentation ℤ_[p] G))} (hiπ : Function.Exact ⇑i ⇑π) (hi : Function.Injective ⇑i) (hπ : Function.Surjective ⇑π) {φ : (ι → MonoidAlgebra ℤ_[p] G) →ₗ[MonoidAlgebra ℤ_[p] G] Y} (hφ : Function.Surjective ⇑φ) {g : ι → G} (hg : Subgroup.closure (Set.range g) = ⊤) : ∃ (β : ↥(MonoidAlgebra.relationModule ℤ_[p] G g) →ₗ[MonoidAlgebra ℤ_[p] G] A), Function.Surjective ⇑β ∧ Nonempty (↥β.ker ≃ₗ[MonoidAlgebra ℤ_[p] G] ↥φ.ker)

The relation module maps onto the kernel of an extension of the augmentation ideal. Let 0 → A → Y → I_G → 0 be an exact sequence of ℤ_p[G]-modules, for I_G the augmentation ideal of ℤ_p[G], and let φ : ℤ_p[G]^ι → Y be onto. For every generating family g : ι → G, there is a surjection from the relation module of g onto A whose kernel is isomorphic to the kernel of φ.

theorem TauCeti.nonempty_ker_linearEquiv_of_relationModule_surjective {p : ℕ} [Fact (Nat.Prime p)] {G : Type u_1} [Group G] [Finite G] {ι : Type u_2} [Fintype ι] {A : Type u_3} [AddCommGroup A] [Module ℤ_[p] A] [Module (MonoidAlgebra ℤ_[p] G) A] [IsScalarTower ℤ_[p] (MonoidAlgebra ℤ_[p] G) A] {g : ι → G} (hg : Subgroup.closure (Set.range g) = ⊤) {n : ℕ} (hι : Fintype.card ι = n + 2) {β : ↥(MonoidAlgebra.relationModule ℤ_[p] G g) →ₗ[MonoidAlgebra ℤ_[p] G] A} (hβ : Function.Surjective ⇑β) [Module.Projective (MonoidAlgebra ℤ_[p] G) ↥β.ker] (hA : Nonempty (TensorProduct ℤ_[p] A ℚ_[p] ≃ₗ[MonoidAlgebra ℤ_[p] G] TensorProduct ℤ_[p] ((Fin n → MonoidAlgebra ℤ_[p] G) × MonoidAlgebra ℤ_[p] G ⧸ RingHom.ker (MonoidAlgebra.augmentation ℤ_[p] G)) ℚ_[p])) : Nonempty (↥β.ker ≃ₗ[MonoidAlgebra ℤ_[p] G] MonoidAlgebra ℤ_[p] G)

A projective kernel of a relation-module surjection is free of rank one. Let G be a finite group, g a generating family of G with n + 2 members, and β a surjection from the relation module of g onto a ℤ_p[G]-module A whose rationalization is (ℤ_p[G]^n × ℤ_p[G] ⧸ I_G) ⊗ ℚ_p, that is, ℚ_p[G]^n ⊕ ℚ_p. If the kernel of β is projective, then it is isomorphic to ℤ_p[G].