The localization of idele cohomology at the places of K #
Let K be a number field, v a finite place of K with completion K_v, and G_K, G_{K_v}
the absolute Galois groups. A K-embedding τ : Kˢ → K_vˢ of separable closures picks out, for
every finite subextension E of Kˢ/K, one place of E above v, together with an
embedding of its completion into K_vˢ. Reading the component of an idele of E at that place
in K_vˢ gives the coordinate at v of the ideles of Kˢ,
ideleCoeffComponent τ : I_{Kˢ} → (K_vˢ)ˣ,
which is equivariant along the decomposition map absoluteGaloisGroupMap τ : G_{K_v} → G_K
(ideleCoeffComponent_smul) and restricts to τ on the principal ideles
(ideleCoeffComponent_principalIdele). It is built without topology on K_vˢ: the components above
v of an idele of E form a unit of K_v ⊗[K] E (TauCeti.finiteAdeleSemilocalHom), which
a ⊗ x ↦ a τ(x) maps to K_vˢ.
Pulling back along this compatible pair is the localization of idele cohomology at v,
ideleBrLocalization v : H²(G_K, I_{Kˢ}) → Br K_v,
the component at v of a global idele class. It does not depend on τ
(ideleBrLocalization_apply): changing τ by h ∈ G_K changes the pair by the inner
automorphism of h, which acts trivially on cohomology. On the image of
H²(G_K, (Kˢ)ˣ) = Br K under the principal ideles it is the localization Br K → Br K_v of
global Brauer classes (ideleBrLocalization_principalIdele). As for a global Brauer class, only
finitely many of the localizations of a class of H²(G_K, I_{Kˢ}) are nonzero
(finite_setOfPred_ideleBrLocalization_ne_zero), so the local invariants of an idele class have
a finite sum. Together, the localizations at all places form the localization map
ideleLocalization K : H²(G_K, I_{Kˢ}) → ⨁_v Br K_v, which on principal idele classes is the
localization map brLocalization K of global Brauer classes
(ideleLocalization_principalIdele).
The same construction works at an infinite place w of K, with completion K_w = ℝ or ℂ:
the components above w of an idele of E form a unit of K_w ⊗[K] E
(TauCeti.GlobalNumberFields.infiniteAdeleSemilocalHom), and an embedding
τ : Kˢ → K_wˢ gives the coordinate ideleCoeffInfiniteComponent τ : I_{Kˢ} → (K_wˢ)ˣ and the
localization
ideleInfiniteBrLocalization w : H²(G_K, I_{Kˢ}) → Br K_w,
with the same properties. Both are instances of one construction from the semi-local components
of adeles, which only uses that these components are natural in E and send x ∈ E to 1 ⊗ x.
Main definitions #
TauCeti.ClassFieldTheory.ideleCoeffComponent τ: the coordinate atvof the ideles ofKˢalongτ.TauCeti.ClassFieldTheory.ideleBrLocalization v: the localizationH²(G_K, I_{Kˢ}) → Br K_v.TauCeti.ClassFieldTheory.ideleCoeffInfiniteComponent τ,TauCeti.ClassFieldTheory.ideleInfiniteBrLocalization w: the coordinate and the localization at an infinite placew.TauCeti.ClassFieldTheory.ideleLocalization K: the localization mapH²(G_K, I_{Kˢ}) → ⨁_v Br K_vat all places.
Main results #
TauCeti.ClassFieldTheory.ideleCoeffComponent_smul: the coordinate is equivariant alongG_{K_v} → G_K.TauCeti.ClassFieldTheory.ideleCoeffComponent_comp: changingτbyh ∈ G_Kprecomposes the coordinate with the action ofh.TauCeti.ClassFieldTheory.ideleCoeffComponent_principalIdele: on principal ideles the coordinate isτ.TauCeti.ClassFieldTheory.ideleBrLocalization_apply: the localization is the pullback along the pair of anyτ.TauCeti.ClassFieldTheory.ideleBrLocalization_principalIdele: on principal idele classes the localization isbrBaseChange K K_v.TauCeti.ClassFieldTheory.ideleInfiniteBrLocalization_apply,TauCeti.ClassFieldTheory.ideleInfiniteBrLocalization_principalIdele: the same two statements at an infinite place.toMul_ideleCoeffComponent_ideleCoeffOfIntermediateField_eq_map_ideleFiniteCoord: forτextending an embedding of a completionE_wabovev, the coordinate atvof an idele ofEis its component atw.toMul_ideleCoeffInfiniteComponent_ideleCoeffOfIntermediateField_eq_map_ideleInfiniteCoord: the corresponding equation at an infinite place.exists_ideleCoeffComponent_ideleCoeffOfIntermediateField_eq,exists_ideleCoeffInfiniteComponent_ideleCoeffOfIntermediateField_eq: the embeddings reading a chosen finite-place or infinite-place coordinate can be chosen simultaneously for all ideles of a finite subextension.exists_ideleCoeffComponent_ideleExtension_eq,exists_ideleCoeffInfiniteComponent_ideleExtension_eq: at every place, some embedding reads the component of each idele ofKas the coordinate of its extension toKˢ.TauCeti.ClassFieldTheory.finite_setOfPred_ideleBrLocalization_ne_zero: a class has nonzero localization at only finitely many finite places.TauCeti.ClassFieldTheory.ideleLocalization_principalIdele: on principal idele classes the localization map isbrLocalization K.
References #
- J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields, 2nd ed., (8.1.17).
- J. Tate, Global class field theory, in J. W. S. Cassels and A. Fröhlich (eds.), Algebraic Number Theory, Chapter VII, §11.
Semi-local components of adeles #
The construction is the same at finite and at infinite places: all it uses is the family of
semi-local components 𝔸_E → F ⊗[K] E of the adeles of the finite Galois subextensions E of
Kˢ/K, for the completion F of K at the place, natural in E.
The localization at a finite place #
The coordinate at v of the ideles of Kˢ along a K-embedding
τ : Kˢ →ₐ[K] K_vˢ of separable closures: an idele a of a finite subextension E of
Kˢ/K goes to the image of its components above v, read in K_v ⊗[K] E, under the
K_v-algebra map a ⊗ x ↦ a τ(x) to K_vˢ
(toMul_ideleCoeffComponent_ideleCoeffOfIntermediateField). A ring
homomorphism from K_v ⊗[K] E ≃ ∏_{w ∣ v} E_w to a field factors through a single factor E_w,
so this reads the component of the idele at one place w of E above v, embedded in
K_vˢ.
Instances For
The coordinate at v of an idele of E: the image of its components above v, read in
K_v ⊗[K] E, under a ⊗ x ↦ a τ(x).
Changing the embedding by an automorphism h of Kˢ precomposes the coordinate at v
with the action of h on the ideles.
The coordinate at v is equivariant along the decomposition map
absoluteGaloisGroupMap τ : G_{K_v} → G_K.
On principal ideles the coordinate at v is τ: the coordinate of the principal idele
of a unit x of Kˢ is τ x.
The localization of idele cohomology at a finite place v,
H²(G_K, I_{Kˢ}) → Br K_v: the pullback along the decomposition map
G_{K_v} → G_K and the coordinate at v of the ideles (ideleCoeffComponent) of an embedding of
separable closures Kˢ → K_vˢ, followed by the identification of H²(G_{K_v}, (K_vˢ)ˣ) with
Br K_v. It does not depend on the embedding (ideleBrLocalization_apply).
Instances For
The localization of idele cohomology is the pullback along any embedding τ of separable
closures, through the decomposition map of τ and the coordinate at v along τ.
On principal idele classes, the localization is the localization of Brauer classes: the
class in H²(G_K, I_{Kˢ}) of a Brauer class x ∈ Br K = H²(G_K, (Kˢ)ˣ), along the principal
ideles, localizes at v to the base change of x to K_v.
The localization at an infinite place #
The coordinate at an infinite place w of the ideles of Kˢ along a K-embedding
τ : Kˢ →ₐ[K] K_wˢ of separable closures: an idele a of a finite subextension E of
Kˢ/K goes to the image of its components above w, read in K_w ⊗[K] E, under the
K_w-algebra map a ⊗ x ↦ a τ(x) to K_wˢ
(toMul_ideleCoeffInfiniteComponent_ideleCoeffOfIntermediateField).
It reads the component of the idele at one place of E above w, embedded in K_wˢ.
Instances For
The coordinate at w of an idele of E: the image of its components above w, read in
K_w ⊗[K] E, under a ⊗ x ↦ a τ(x).
The coordinate at an infinite place through a completion of a finite subextension.
If τ agrees on E with an embedding τ' of E_u, the coordinate is the component at u
embedded by τ'. No normality of E/K is needed.
Every infinite completion coordinate of an idele of a finite subextension is a coordinate
of the corresponding idele of Kˢ, along suitable embeddings of separable closures: the
archimedean analogue of exists_ideleCoeffComponent_ideleCoeffOfIntermediateField_eq. The
subextension need not be normal over K, and the embeddings work simultaneously for all its
ideles.
Changing the embedding by an automorphism h of Kˢ precomposes the coordinate at w
with the action of h on the ideles.
The coordinate at w is equivariant along the decomposition map
absoluteGaloisGroupMap τ : G_{K_w} → G_K.
On principal ideles the coordinate at w is τ: the coordinate of the principal idele
of a unit x of Kˢ is τ x.
The localization of idele cohomology at an infinite place w,
H²(G_K, I_{Kˢ}) → Br K_w: the pullback along the decomposition map G_{K_w} → G_K and the
coordinate at w of the ideles (ideleCoeffInfiniteComponent) of an embedding of separable
closures Kˢ → K_wˢ, followed by the identification of H²(G_{K_w}, (K_wˢ)ˣ) with Br K_w. It
does not depend on the embedding (ideleInfiniteBrLocalization_apply).
Instances For
The localization of idele cohomology at w is the pullback along any embedding τ of
separable closures, through the decomposition map of τ and the coordinate at w along τ.
On principal idele classes, the localization at w is the localization of Brauer
classes: the class in H²(G_K, I_{Kˢ}) of a Brauer class x ∈ Br K, along the principal
ideles, localizes at w to the base change of x to K_w.
Finite support #
A class of H²(G_K, I_{Kˢ}) is represented by a cocycle with finitely many values, all ideles of
one finite Galois subextension E of Kˢ/K. At a place w of E that is unramified over K and
at which all these ideles are units, choose the embedding τ of separable closures through the
completion E_w: the coordinate along τ is then the component at w
(toMul_ideleCoeffComponent_ideleCoeffOfIntermediateField_eq_map_ideleFiniteCoord), so the
localization at the place v below w is inflated from a cocycle of the unramified layer
E_w/K_v with values in its units of valuation one, and vanishes
(mk_eq_zero_of_forall_mem_unitFiltration_zero).
The coordinate at v through a completion of E. If the embedding τ : Kˢ → K_vˢ agrees
on a finite subextension E with an embedding τ' of the completion E_w at a place w
of E above v, then the coordinate at v along τ of an idele of E is its component at w,
embedded in K_vˢ by τ'.
Every finite completion coordinate of an idele of a finite subextension is a coordinate
of the corresponding idele of Kˢ, along suitable embeddings of separable closures. The
subextension need not be normal over K, and the embeddings work simultaneously for all its
ideles.
The localization of idele cohomology is finitely supported: a class of H²(G_K, I_{Kˢ})
has nonzero localization ideleBrLocalization v at only finitely many finite places v.
The coordinates of the ideles of K #
An idele of K is an invariant idele of Kˢ, by extension to the bottom subextension K of
Kˢ/K. At every place, some embedding of separable closures reads its component there as its
coordinate. These are the coordinates consumed by the character formula for the Artin map on
idele classes of K.
The coordinates at a finite place of the ideles of K. Some embedding
τ : Kˢ → K_vˢ of separable closures reads, for every idele a of K, its component
a_v ∈ K_vˣ as the coordinate at v of the extension of a to Kˢ.
The coordinates at an infinite place of the ideles of K. Some embedding
τ : Kˢ → K_wˢ of separable closures reads, for every idele a of K, its component
a_w ∈ K_wˣ as the coordinate at w of the extension of a to Kˢ.
The localization map at all places #
The localization of idele cohomology at all places, H²(G_K, I_{Kˢ}) → ⨁_v Br K_v: the
localizations ideleBrLocalization v at the finite places, a finitely supported family by
finite_setOfPred_ideleBrLocalization_ne_zero, together with the localizations
ideleInfiniteBrLocalization w at the infinite places. Its target is the target of the
localization map brLocalization K of global Brauer classes, which it extends along the principal
ideles (ideleLocalization_principalIdele).
Instances For
The component of the localization map at a finite place v is ideleBrLocalization v.
The component of the localization map at an infinite place w is
ideleInfiniteBrLocalization w.
On principal idele classes, the localization map is that of Brauer classes: the class in
H²(G_K, I_{Kˢ}) of a Brauer class x ∈ Br K, along the principal ideles, localizes to the
family brLocalization K x of its base changes to all completions.