Every layer of the idele-class formation as a Galois module #
Let K be a number field and V ◁ U a finite normal layer of the idele-class formation
globalFormation K, whose ground subgroup U is any open subgroup of G_K. Let E be the fixed
field of V, a finite subextension of Kˢ/K, let F be a field over which E is an algebra,
and let e : U ⧸ V ≃* Gal(E/F) send the class of each u ∈ U to the restriction of u
to E. Such an e is unique when it exists, and it exists when F is the fixed field of U, by
the Galois correspondence for Kˢ/K.
The coefficient module (C_{Kˢ})^V of the layer is the idele class group C_E of E, through
ideleClassLevelEquiv. This file proves that, along e, this identification intertwines the
action of U ⧸ V on the layer with the Galois action of Gal(E/F) on C_E, the module of
TauCeti.ClassFieldTheory.ideleClassRep F E (NormalLayer.ideleClassLevelEquiv_ρ).
Consequently the cohomology of the layer is the Galois cohomology Hⁿ(Gal(E/F), C_E) in every
degree (NormalLayer.ideleClassCohomologyEquiv), its Tate cohomology is the Tate cohomology
Hⁿ_T(Gal(E/F), C_E) in every integer degree (NormalLayer.ideleClassTateCohomologyIso), and
its Herbrand quotient is h(C_E), computed for the action of Gal(E/F)
(NormalLayer.herbrandQuotient_rep_globalFormation).
For a finite Galois extension L/K given abstractly, the layer of fixingOpenNormalSubgroup K L
has fixed field the image E of a K-embedding ι : L →ₐ[K] Kˢ, and C_E is C_L transported
along ι. A K-isomorphism of extensions identifies the cohomology of their idele classes
(ideleClassCohomologyCongr), so the cohomology of the layer is Hⁿ(Gal(L/K), C_L)
(globalLayerCohomologyEquiv). This is the form in which vanishing statements on the layers of
globalFormation K, such as that of H¹, apply to the idele classes of a finite Galois extension.
The reductions of TauCeti.NumberTheory.ClassFieldTheory.Formation.PrimeDegree derive the
vanishing of H¹ and the bound #H² ∣ [U : V] for a layer from the two fundamental inequalities
on all the layers of prime degree inside it, and the ground subgroups of those layers are proper
open subgroups of G_K in general. The Herbrand quotient of each such layer is the Herbrand
quotient of the idele classes of a cyclic extension E/F of number fields, which
TauCeti.ClassFieldTheory.herbrandQuotient_ideleClassRep reduces to the S-ideles and the
S-units of E.
Main results #
TauCeti.ClassFieldTheory.NormalLayer.ideleClassLevelEquiv_ρ: the identification ofC_Ewith the coefficient module of the layer is equivariant alonge.TauCeti.ClassFieldTheory.NormalLayer.isIntertwiningMap_ideleClassLevelEquiv: the same statement in the form of a compatible pair, as taken by the change-of-group maps of cohomology.TauCeti.ClassFieldTheory.NormalLayer.ideleClassCohomologyEquiv: the cohomology of the layer isHⁿ(Gal(E/F), C_E).TauCeti.ClassFieldTheory.NormalLayer.ideleClassTateCohomologyIso: the Tate cohomology of the layer isHⁿ_T(Gal(E/F), C_E).TauCeti.ClassFieldTheory.NormalLayer.herbrandQuotient_rep_globalFormation: the Herbrand quotient of the layer ish(C_E).TauCeti.ClassFieldTheory.ideleClassCohomologyCongr: aK-isomorphismL ≃ₐ[K] MidentifiesHⁿ(Aut(L/K), C_L)withHⁿ(Aut(M/K), C_M).TauCeti.ClassFieldTheory.globalLayerCohomologyEquiv: the cohomology of the layer of a finite Galois extensionL/KisHⁿ(Gal(L/K), C_L).TauCeti.ClassFieldTheory.subsingleton_groupCohomology_ideleClassRep_iff_subsingleton_H:Hⁿ(Gal(L/K), C_L)vanishes exactly when the cohomology of the layer does.
References #
- J. S. Milne, Class Field Theory, v4.03, Chapter VII, §§4–5.
- J. Neukirch, Algebraic Number Theory, Chapter VI, §§2 and 4.
The coefficient module #
The coefficient module of a layer of the idele-class formation is C_E as a Galois
module: if E is the fixed field of the top subgroup of the layer and e : U ⧸ V ≃* Gal(E/F)
sends the class of each u to its restriction to E, then τ ∈ Gal(E/F) acts on C_E as
e⁻¹ τ acts on the coefficient module (C_{Kˢ})^V of the layer, through ideleClassLevelEquiv.
ideleClassLevelEquiv and the inverse of e form a compatible pair, in the form taken by the
change-of-group maps of Tate cohomology.
Cohomology and the Herbrand quotient #
The cohomology of a layer of the idele-class formation is the Galois cohomology of the
idele classes: if E is the fixed field of the top subgroup of the layer and
e : U ⧸ V ≃* Gal(E/F) sends the class of each u to its restriction to E, then
Hⁿ(U ⧸ V, (C_{Kˢ})^V) ≃ Hⁿ(Gal(E/F), C_E) in every degree n.
Instances For
ideleClassCohomologyEquiv is the inverse of the isomorphism groupCohomology.mapIso along
e⁻¹ and ideleClassLevelEquiv.
The inverse of ideleClassCohomologyEquiv is the isomorphism groupCohomology.mapIso along
e⁻¹ and ideleClassLevelEquiv.
The Tate cohomology of a layer of the idele-class formation is the Tate cohomology of the
idele classes: if E is the fixed field of the top subgroup of the layer and
e : U ⧸ V ≃* Gal(E/F) sends the class of each u to its restriction to E, then
Hⁿ_T(U ⧸ V, (C_{Kˢ})^V) ≅ Hⁿ_T(Gal(E/F), C_E) in every integer degree n.
Instances For
The forward map of ideleClassTateCohomologyIso is the inverse of the isomorphism
TateCohomology.mapIso along e⁻¹ and ideleClassLevelEquiv.
The inverse of ideleClassTateCohomologyIso is the isomorphism TateCohomology.mapIso along
e⁻¹ and ideleClassLevelEquiv.
The Herbrand quotient of a layer of the idele-class formation is that of the idele
classes: if E is the fixed field of the top subgroup of the layer and e : U ⧸ V ≃* Gal(E/F)
sends the class of each u to its restriction to E, then the coefficient module of the layer has
Herbrand quotient h(C_E), computed for the action of Gal(E/F).
Transport along isomorphisms of extensions #
The cohomology of the idele classes is invariant under isomorphism of extensions: a
K-isomorphism e : L ≃ₐ[K] M identifies Hⁿ(Aut(L/K), C_L) with Hⁿ(Aut(M/K), C_M), along
conjugation by e on the automorphism groups (AlgEquiv.autCongr) and transport of idele classes
on the coefficients (GlobalNumberFields.ideleClassEquiv).
Instances For
ideleClassCohomologyCongr e is the isomorphism groupCohomology.mapIso along e.autCongr
and the transport of idele classes.
The layer of a finite Galois extension #
The cohomology of the layer of a finite Galois extension is the Galois cohomology of its
idele classes: for L/K finite Galois and a K-embedding ι : L →ₐ[K] Kˢ, the cohomology of
the layer of fixingOpenNormalSubgroup K L in the idele-class formation is Hⁿ(Gal(L/K), C_L) in
every degree n. The layer is read on the image E of ι (ideleClassCohomologyEquiv), whose
Galois group is identified with Gal(L/K) by restriction along ι (layerGalEquiv), and C_E
is identified with C_L along ι (ideleClassCohomologyCongr).
Instances For
globalLayerCohomologyEquiv ι reads the layer on the image of ι
(NormalLayer.ideleClassCohomologyEquiv) and transports the result to L along ι
(ideleClassCohomologyCongr).
The idele-class cohomology of a finite Galois extension vanishes with that of its layer:
for L/K finite Galois, Hⁿ(Gal(L/K), C_L) is trivial exactly when Hⁿ of the layer of
fixingOpenNormalSubgroup K L in the idele-class formation is.