The units of an unramified extension have no Tate cohomology

Let L/K be a finite unramified Galois extension of nonarchimedean local fields, with Galois group G. This file proves that the unit group 𝒪[L]ˣ = U(L, 0), with its Galois action and read as an integral representation of G, has vanishing Tate cohomology in every degree.

In degree zero, H-hat^0(G, 𝒪[L]ˣ) = 𝒪[K]ˣ / N_{L/K}(𝒪[L]ˣ) vanishes because the norm of an unramified extension maps 𝒪[L]ˣ onto 𝒪[K]ˣ (TauCeti.map_normUnits_unitFiltration). The group G is cyclic (TauCeti.isCyclic_algEquiv) and the Herbrand quotient of 𝒪[L]ˣ is 1 (TauCeti.TateCohomology.herbrandQuotient_unitFiltration_zero), so H-hat^(-1)(G, 𝒪[L]ˣ) vanishes as well, and two-periodicity gives every other degree.

The statement concerns the whole group G. Since L/E is again unramified for every intermediate field E, the same theorem over E applies to the subgroup Gal(L/E) of G.

This is the local factor at the places outside S in the computation of the Herbrand quotient of the S-ideles of a cyclic extension of number fields: the unit groups at places unramified in L contribute nothing to the Tate cohomology of the ideles.

⚠ For ramified extensions only the Herbrand quotient survives: for L = ℚ_2(√2) the norms of the units of 𝒪[L] have index 2 in ℤ_2ˣ, so H-hat^0(G, 𝒪[L]ˣ) has order 2.

Main results

• TauCeti.TateCohomology.isZero_tateCohomology_unitFiltration_zero_of_isUnramified: H-hat^n(G, 𝒪[L]ˣ) = 0 for every n : ℤ, for an unramified Galois extension L/K.

References

• J.-P. Serre, Local Fields, Chapter V, §2, Proposition 3 and Chapter VIII, §1.
• J. S. Milne, Class Field Theory, Chapter III, Proposition 1.2 and Chapter VII, Lemma 2.4.

theorem TauCeti.TateCohomology.isZero_tateCohomology_unitFiltration_zero_of_isUnramified {K L : Type} [Field K] [ValuativeRel K] [TopologicalSpace K] [IsNonarchimedeanLocalField K] [Field L] [ValuativeRel L] [TopologicalSpace L] [IsNonarchimedeanLocalField L] [Algebra K L] [ValuativeExtension K L] [Module.Finite K L] [IsGalois K L] [IsUnramified K L] (n : ℤ) : CategoryTheory.Limits.IsZero (tateCohomology (Rep.ofMulDistribMulAction Gal(L/K) ↥(unitFiltration L 0)) n)

The units of an unramified extension have no Tate cohomology. For a finite unramified Galois extension L/K of nonarchimedean local fields, the unit group 𝒪[L]ˣ has vanishing Tate cohomology H-hat^n(Gal(L/K), 𝒪[L]ˣ) in every degree n : ℤ.