The third cohomology of the units of a local field #
For a nonarchimedean local field K with absolute Galois group G_K, this file proves
H³(G_K, (Kˢ)ˣ) = 0
on Mathlib's continuous cohomology of the coefficient object TauCeti.ClassFieldTheory.unitsRep K
(TauCeti.ClassFieldTheory.subsingleton_h3_unitsRep). The units formation of K is a class
formation (TauCeti.ClassFieldTheory.localClassFormation), so the continuous third cohomology of
its module vanishes by Tate's theorem on every finite layer and the finite-quotient colimit
(TauCeti.ClassFieldTheory.ClassFormation.subsingleton_continuousCohomology_three). The module of
the units formation is (Kˢ)ˣ over Gal(Kˢ/K), while unitsRep K is the same group over G_K,
on which it acts through the restriction isomorphism
TauCeti.absoluteGaloisGroupRestrictEquiv : G_K ≃ₜ* Gal(Kˢ/K); the two continuous cohomologies are
compared along that isomorphism.
This vanishing is one input of the cohomological dimension cd_ℓ G_K = 2 of a finite extension of
ℚ_p. For a finite extension L/K, the Kummer sequence 1 → μ_ℓ → (Lˢ)ˣ → (Lˢ)ˣ → 1 and the
vanishing of H³(G_L, (Lˢ)ˣ) identify H³(G_L, μ_ℓ) with the cokernel of the ℓ-th power map on
H²(G_L, (Lˢ)ˣ); that cokernel vanishes by the second input, the invariant isomorphism
Br(L) ≅ ℚ/ℤ, which makes H²(G_L, (Lˢ)ˣ) divisible.
Main results #
TauCeti.ClassFieldTheory.subsingleton_h3_unitsRep:H³(G_K, (Kˢ)ˣ) = 0.
References #
- J.-P. Serre, Galois Cohomology, Ch. II, §5.3.
- J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields, 2nd ed., (7.1.8).
H³(G_K, (Kˢ)ˣ) = 0 for a nonarchimedean local field K: the units formation of K is a
class formation, whose continuous third cohomology vanishes by Tate's theorem on every finite layer,
H-hat^3(Gal(L/K), Lˣ) ≅ H-hat^1(Gal(L/K), ℤ) = 0, and the finite-quotient colimit.