The localizations of a relative Brauer class and the cohomology of the ideles

Let L/K be a finite Galois extension of number fields with Galois group G, embedded in Kˢ by σ. A class x ∈ H²(G, Lˣ) gives the Brauer class relBrInfl K L σ x ∈ Br K, whose localization at a place v of K is its base change to the completion K_v. It also gives, through the principal ideles Lˣ → I_L (unitsToIdeles), a class of H²(G, I_L), which has a semi-local component in H²(G, (K_v ⊗[K] L)ˣ) at every place v, finite (idelesToSemilocalUnits) or infinite (idelesToInfiniteSemilocalUnits). This file shows that the two families carry the same information: the semi-local component of x at v vanishes exactly when the localization of its Brauer class at v does. Hence the image of x in H²(G, I_L) vanishes exactly when every localization of its Brauer class vanishes (map_unitsToIdeles_eq_zero_iff_brLocalization_eq_zero).

This is the idelic half of the injectivity of Br K → ⨁_v Br K_v (Albert–Brauer–Hasse–Noether): a split class with trivial localizations dies in H²(G, I_L), so by the exact sequence 1 → Lˣ → I_L → C_L → 1 it comes from H¹(G, C_L).

The comparison at a place v goes through Shapiro's lemma. For a place w of L above v, the semi-local units are coinduced from the units of the completion L_w as a representation of the decomposition group D_w (semilocalUnitsCoindIso, infiniteSemilocalUnitsCoindIso), so the semi-local component of x is determined by its restriction to D_w, read in L_wˣ (groupCohomology.map_id_eq_zero_iff_of_comp_eq_resCoindToHom). The decomposition group is the Galois group of L_w/K_v (decompositionHom), and there the restriction is the map H²(Gal(L/K), Lˣ) → H²(Gal(L_w/K_v), L_wˣ) along restriction of automorphisms and the inclusion Lˣ → L_wˣ (map_unitsToIdeles_idelesToSemilocalUnits_eq_zero_iff). That map is how base change to K_v acts on inflated Brauer classes (brBaseChange_relBrInfl), and inflation into Br K_v is injective (relBrInfl_injective).

Main results

• TauCeti.ClassFieldTheory.map_unitsToIdeles_idelesToSemilocalUnits_eq_zero_iff, TauCeti.ClassFieldTheory.map_unitsToIdeles_idelesToInfiniteSemilocalUnits_eq_zero_iff: in every degree, the semi-local component of a class of Hⁿ(G, Lˣ) at a finite or infinite place v vanishes exactly when its image in Hⁿ(Gal(L_w/K_v), L_wˣ) does.
• map_unitsToIdeles_idelesToSemilocalUnits_eq_zero_iff_brBaseChange_eq_zero, map_unitsToIdeles_idelesToInfiniteSemilocalUnits_eq_zero_iff_brBaseChange_eq_zero: in degree two, it vanishes exactly when the localization at v of the Brauer class does.
• TauCeti.ClassFieldTheory.map_unitsToIdeles_eq_zero_iff_brLocalization_eq_zero: the image of x ∈ H²(G, Lˣ) in H²(G, I_L) vanishes exactly when every localization of relBrInfl K L σ x vanishes.

References

• J. S. Milne, Class Field Theory, Chapter VII, Lemma 2.4 and Proposition 2.5, and Chapter VIII, §4.
• J. Tate, Global class field theory, in J. W. S. Cassels and A. Fröhlich (eds.), Algebraic Number Theory, Chapter VII, §§6–7.

theorem TauCeti.ClassFieldTheory.map_unitsToIdeles_idelesToSemilocalUnits_eq_zero_iff {K : Type} [Field K] [NumberField K] {L : Type} [Field L] [NumberField L] [Algebra K L] (v : IsDedekindDomain.HeightOneSpectrum (NumberField.RingOfIntegers K)) (w : IsDedekindDomain.HeightOneSpectrum (NumberField.RingOfIntegers L)) [w.asIdeal.LiesOver v.asIdeal] [IsGalois K L] (n : ℕ) (x : ↑(groupCohomology (Rep.ofMulDistribMulAction Gal(L/K) Lˣ) n)) : (CategoryTheory.ConcreteCategory.hom (groupCohomology.map (MonoidHom.id Gal(L/K)) (CategoryTheory.CategoryStruct.comp (unitsToIdeles K L) (idelesToSemilocalUnits L v)) n)) x = 0 ↔ (CategoryTheory.ConcreteCategory.hom (groupCohomology.map ((AlgEquiv.restrictNormalHom L).comp (AlgEquiv.restrictScalarsHom K)) (unitsBaseChangeHom K L (IsDedekindDomain.HeightOneSpectrum.adicCompletion K v) (IsDedekindDomain.HeightOneSpectrum.adicCompletion L w)) n)) x = 0

The semi-local component of a class of Hⁿ(Gal(L/K), Lˣ) at a finite place v vanishes exactly when the class vanishes in Hⁿ(Gal(L_w/K_v), L_wˣ), for any place w of L above v, along restriction of automorphisms and the inclusion Lˣ → L_wˣ.

theorem TauCeti.ClassFieldTheory.map_unitsToIdeles_idelesToSemilocalUnits_eq_zero_iff_brBaseChange_eq_zero {K : Type} [Field K] [NumberField K] {L : Type} [Field L] [NumberField L] [Algebra K L] {v : IsDedekindDomain.HeightOneSpectrum (NumberField.RingOfIntegers K)} [IsGalois K L] (σ : L →ₐ[K] SeparableClosure K) (x : ↑(groupCohomology (Rep.ofMulDistribMulAction Gal(L/K) Lˣ) 2)) : (CategoryTheory.ConcreteCategory.hom (groupCohomology.map (MonoidHom.id Gal(L/K)) (CategoryTheory.CategoryStruct.comp (unitsToIdeles K L) (idelesToSemilocalUnits L v)) 2)) x = 0 ↔ (brBaseChange K (IsDedekindDomain.HeightOneSpectrum.adicCompletion K v)) ((relBrInfl K L σ) x) = 0

The semi-local component of a class of H²(Gal(L/K), Lˣ) at a finite place v vanishes exactly when the localization at v of its Brauer class vanishes.

theorem TauCeti.ClassFieldTheory.map_unitsToIdeles_idelesToInfiniteSemilocalUnits_eq_zero_iff {K : Type} [Field K] [NumberField K] {L : Type} [Field L] [NumberField L] [Algebra K L] (v : NumberField.InfinitePlace K) (w : NumberField.InfinitePlace L) [w.LiesOver v] [IsGalois K L] (n : ℕ) (x : ↑(groupCohomology (Rep.ofMulDistribMulAction Gal(L/K) Lˣ) n)) : (CategoryTheory.ConcreteCategory.hom (groupCohomology.map (MonoidHom.id Gal(L/K)) (CategoryTheory.CategoryStruct.comp (unitsToIdeles K L) (idelesToInfiniteSemilocalUnits L v)) n)) x = 0 ↔ (CategoryTheory.ConcreteCategory.hom (groupCohomology.map ((AlgEquiv.restrictNormalHom L).comp (AlgEquiv.restrictScalarsHom K)) (unitsBaseChangeHom K L v.Completion w.Completion) n)) x = 0

The semi-local component of a class of Hⁿ(Gal(L/K), Lˣ) at an infinite place v vanishes exactly when the class vanishes in Hⁿ(Gal(L_w/K_v), L_wˣ), for any place w of L above v, along restriction of automorphisms and the inclusion Lˣ → L_wˣ.

theorem TauCeti.ClassFieldTheory.map_unitsToIdeles_idelesToInfiniteSemilocalUnits_eq_zero_iff_brBaseChange_eq_zero {K : Type} [Field K] [NumberField K] {L : Type} [Field L] [NumberField L] [Algebra K L] {v : NumberField.InfinitePlace K} [IsGalois K L] (σ : L →ₐ[K] SeparableClosure K) (x : ↑(groupCohomology (Rep.ofMulDistribMulAction Gal(L/K) Lˣ) 2)) : (CategoryTheory.ConcreteCategory.hom (groupCohomology.map (MonoidHom.id Gal(L/K)) (CategoryTheory.CategoryStruct.comp (unitsToIdeles K L) (idelesToInfiniteSemilocalUnits L v)) 2)) x = 0 ↔ (brBaseChange K v.Completion) ((relBrInfl K L σ) x) = 0

The semi-local component of a class of H²(Gal(L/K), Lˣ) at an infinite place v vanishes exactly when the localization at v of its Brauer class vanishes.

All places together

theorem TauCeti.ClassFieldTheory.map_unitsToIdeles_eq_zero_iff_brLocalization_eq_zero {K : Type} [Field K] [NumberField K] {L : Type} [Field L] [NumberField L] [Algebra K L] [IsGalois K L] (σ : L →ₐ[K] SeparableClosure K) (x : ↑(groupCohomology (Rep.ofMulDistribMulAction Gal(L/K) Lˣ) 2)) : (CategoryTheory.ConcreteCategory.hom (groupCohomology.map (MonoidHom.id Gal(L/K)) (unitsToIdeles K L) 2)) x = 0 ↔ (brLocalization K) ((relBrInfl K L σ) x) = 0

A split Brauer class has trivial localizations exactly when it dies in the cohomology of the ideles. For L/K finite Galois, embedded in Kˢ by σ, a class x ∈ H²(Gal(L/K), Lˣ) maps to zero in H²(Gal(L/K), I_L) under the principal ideles exactly when every localization of the Brauer class relBrInfl K L σ x, at the finite and at the infinite places, vanishes.