Tate cohomology of a complete filtered representation

Let M be a representation of a finite group G, and let F 0 = ⊤, F 1, F 2, … be G-stable submodules of M that are separated, meaning ⋂ n, F n = 0, and complete, meaning that every series ∑ n, x n with x n ∈ F n has a limit s, in the sense that s - ∑ i < n, x i ∈ F n for every n. Suppose that each F n surjects onto a representation Q n with kernel contained in F (n + 1), and that every Q n has vanishing Tate cohomology in degree 0, respectively -1. Then so does M. In the main case F is a decreasing filtration, the kernel is exactly F (n + 1), and Q n is the graded piece F n / F (n + 1); neither assumption is needed.

The proof is by successive approximation, using the low-degree descriptions H-hat^0(G, M) = Mᴳ / N_G M and H-hat^(-1)(G, M) = ker N_G / I_G M. In degree 0, an invariant m ∈ F n is a norm modulo F (n + 1) because its image in Q n is a norm there; the correction terms y n ∈ F n sum to some s with N_G s = m. In degree -1, an element m of norm zero lies in I_G M modulo F (n + 1) once it lies in F n, and I_G M is the set of sums ∑ g, (g • y g - y g), so one approximates each of the finitely many coordinates y g separately.

The family F is not required to be decreasing, and the quotients Q n are not formed as quotient representations. Instead, each is the target of a surjective morphism π n from the subrepresentation F n, whose kernel is contained in F (n + 1). When F is decreasing, Q n is thus a quotient of F n refining the graded piece F n / F (n + 1); this is how the graded pieces arise in practice, with kernel exactly F (n + 1), but only the containment is needed. Vanishing in degrees 0 and -1 on every Q n gives M Herbrand quotient 1. For finite cyclic G, Tate cohomology is two-periodic, so it also makes M cohomologically trivial.

The motivating application is to the units of a finite Galois extension L/K of nonarchimedean local fields. For a uniformizer ϖ of K and a Galois-stable lattice A ⊆ 𝒪[L], free over 𝒪[K][G], with A * A ≤ ϖ • A (TauCeti.exists_span_orbit_mul_le_smul), the subgroups 1 + ϖ ^ (n + 1) • A filter an open subgroup of the units of 𝒪[L], and each graded piece is A / ϖ • A, free over (𝒪[K] ⧸ ϖ)[G].

Main results

• TauCeti.TateCohomology.isZero_tateCohomology_zero_of_filtration: degree-zero Tate cohomology vanishes if it vanishes on every Q n.
• TauCeti.TateCohomology.isZero_tateCohomology_negOne_of_filtration: degree -1 Tate cohomology vanishes if it vanishes on every Q n.
• TauCeti.TateCohomology.isZero_tateCohomology_of_filtration: for finite cyclic G, all Tate cohomology vanishes if it vanishes on every Q n in degrees 0 and -1.
• TauCeti.TateCohomology.herbrandQuotient_eq_one_of_filtration: the Herbrand quotient is 1 if every Q n has vanishing Tate cohomology in degrees 0 and -1.

References

• J.-P. Serre, Local Fields, Chapter IX, §3, Lemma 2 (the cohomology of a complete filtered module) and Proposition 3 (the units of a finite Galois extension of local fields).
• J. Neukirch, Algebraic Number Theory, Chapter V, Proposition 1.2.

theorem TauCeti.TateCohomology.isZero_tateCohomology_zero_of_filtration {R G : Type u} [CommRing R] [Group G] [Fintype G] {M : Rep.{u, u, u} R G} {F : ℕ → Submodule R ↑M} {Q : ℕ → Rep.{u, u, u} R G} (hF : ∀ (n : ℕ) (g : G), F n ≤ Submodule.comap (M.ρ g) (F n)) (π : (n : ℕ) → M.subrepresentation (F n) ⋯ ⟶ Q n) (hF0 : F 0 = ⊤) (hsep : ∀ (x : ↑M), (∀ (n : ℕ), x ∈ F n) → x = 0) (hcomplete : ∀ (x : ℕ → ↑M), (∀ (n : ℕ), x n ∈ F n) → ∃ (s : ↑M), ∀ (n : ℕ), s - ∑ i ∈ Finset.range n, x i ∈ F n) (hπ : ∀ (n : ℕ), Function.Surjective ⇑(Rep.Hom.hom (π n))) (hker : ∀ (n : ℕ) (x : ↥(F n)), (Rep.Hom.hom (π n)) x = 0 → ↑x ∈ F (n + 1)) (hQ : ∀ (n : ℕ), CategoryTheory.Limits.IsZero (tateCohomology (Q n) 0)) : CategoryTheory.Limits.IsZero (tateCohomology M 0)

Degree-zero Tate cohomology of a complete filtered representation. Let F be a separated and complete family of G-stable submodules of M with F 0 = ⊤, and let π n be a surjection from F n onto Q n whose kernel lies in F (n + 1). If every Q n has vanishing degree-zero Tate cohomology, so does M: every invariant of M is a norm.

theorem TauCeti.TateCohomology.isZero_tateCohomology_negOne_of_filtration {R G : Type u} [CommRing R] [Group G] [Fintype G] {M : Rep.{u, u, u} R G} {F : ℕ → Submodule R ↑M} {Q : ℕ → Rep.{u, u, u} R G} (hF : ∀ (n : ℕ) (g : G), F n ≤ Submodule.comap (M.ρ g) (F n)) (π : (n : ℕ) → M.subrepresentation (F n) ⋯ ⟶ Q n) (hF0 : F 0 = ⊤) (hsep : ∀ (x : ↑M), (∀ (n : ℕ), x ∈ F n) → x = 0) (hcomplete : ∀ (x : ℕ → ↑M), (∀ (n : ℕ), x n ∈ F n) → ∃ (s : ↑M), ∀ (n : ℕ), s - ∑ i ∈ Finset.range n, x i ∈ F n) (hπ : ∀ (n : ℕ), Function.Surjective ⇑(Rep.Hom.hom (π n))) (hker : ∀ (n : ℕ) (x : ↥(F n)), (Rep.Hom.hom (π n)) x = 0 → ↑x ∈ F (n + 1)) (hQ : ∀ (n : ℕ), CategoryTheory.Limits.IsZero (tateCohomology (Q n) (-1))) : CategoryTheory.Limits.IsZero (tateCohomology M (-1))

Degree -1 Tate cohomology of a complete filtered representation. Let F be a separated and complete family of G-stable submodules of M with F 0 = ⊤, and let π n be a surjection from F n onto Q n whose kernel lies in F (n + 1). If every Q n has vanishing degree -1 Tate cohomology, so does M: every element of M of norm zero lies in the coinvariant kernel I_G M.

theorem TauCeti.TateCohomology.isZero_tateCohomology_of_filtration {R G : Type u} [CommRing R] [Group G] [Fintype G] {M : Rep.{u, u, u} R G} {F : ℕ → Submodule R ↑M} {Q : ℕ → Rep.{u, u, u} R G} (hF : ∀ (n : ℕ) (g : G), F n ≤ Submodule.comap (M.ρ g) (F n)) (π : (n : ℕ) → M.subrepresentation (F n) ⋯ ⟶ Q n) [IsCyclic G] (hF0 : F 0 = ⊤) (hsep : ∀ (x : ↑M), (∀ (n : ℕ), x ∈ F n) → x = 0) (hcomplete : ∀ (x : ℕ → ↑M), (∀ (n : ℕ), x n ∈ F n) → ∃ (s : ↑M), ∀ (n : ℕ), s - ∑ i ∈ Finset.range n, x i ∈ F n) (hπ : ∀ (n : ℕ), Function.Surjective ⇑(Rep.Hom.hom (π n))) (hker : ∀ (n : ℕ) (x : ↥(F n)), (Rep.Hom.hom (π n)) x = 0 → ↑x ∈ F (n + 1)) (hQ₀ : ∀ (n : ℕ), CategoryTheory.Limits.IsZero (tateCohomology (Q n) 0)) (hQ₁ : ∀ (n : ℕ), CategoryTheory.Limits.IsZero (tateCohomology (Q n) (-1))) (k : ℤ) : CategoryTheory.Limits.IsZero (tateCohomology M k)

A complete filtered representation of a finite cyclic group is cohomologically trivial if the quotients Q n of the steps F n are. Let F be a separated and complete family of G-stable submodules of M with F 0 = ⊤, and let π n be a surjection from F n onto Q n whose kernel lies in F (n + 1). If every Q n has vanishing Tate cohomology in degrees 0 and -1, then by two-periodicity M has vanishing Tate cohomology in every degree.

theorem TauCeti.TateCohomology.herbrandQuotient_eq_one_of_filtration {R G : Type u} [CommRing R] [Group G] [Fintype G] {M : Rep.{u, u, u} R G} {F : ℕ → Submodule R ↑M} {Q : ℕ → Rep.{u, u, u} R G} (hF : ∀ (n : ℕ) (g : G), F n ≤ Submodule.comap (M.ρ g) (F n)) (π : (n : ℕ) → M.subrepresentation (F n) ⋯ ⟶ Q n) (hF0 : F 0 = ⊤) (hsep : ∀ (x : ↑M), (∀ (n : ℕ), x ∈ F n) → x = 0) (hcomplete : ∀ (x : ℕ → ↑M), (∀ (n : ℕ), x n ∈ F n) → ∃ (s : ↑M), ∀ (n : ℕ), s - ∑ i ∈ Finset.range n, x i ∈ F n) (hπ : ∀ (n : ℕ), Function.Surjective ⇑(Rep.Hom.hom (π n))) (hker : ∀ (n : ℕ) (x : ↥(F n)), (Rep.Hom.hom (π n)) x = 0 → ↑x ∈ F (n + 1)) (hQ₀ : ∀ (n : ℕ), CategoryTheory.Limits.IsZero (tateCohomology (Q n) 0)) (hQ₁ : ∀ (n : ℕ), CategoryTheory.Limits.IsZero (tateCohomology (Q n) (-1))) : herbrandQuotient M = 1

The Herbrand quotient of a complete filtered representation. Under the hypotheses of isZero_tateCohomology_of_filtration, except that G need not be cyclic, the Herbrand quotient of M is 1: it only involves Tate cohomology in degrees 0 and -1.