LocalClassFieldTheory.localBlockFamilyHerbrandH0Equiv
Degree-zero cohomology for a finite family of local blocks.
Source context 8 commands
Surrounding namespaces, section variables, and local commands at this declaration.
noncomputable section
namespace LocalClassFieldTheory
open AlgebraicNumberTheory.Valuations
open HilbertRamification
open CyclicCohomology.ProfiniteCohomology.Herbrand
open CyclicCohomology
universe u v w
variable {K : Type u} {L : Type v}
[Field K] [Field L] [Algebra K L]
[FiniteDimensional K L] [IsGalois K L]Lean source · full declaration
def localBlockFamilyHerbrandH0Equiv
{ι : Type w} [Fintype ι]
(d : ι → LocalPlaceDatum K L)
(σ : L ≃ₐ[K] L)
(hgen : ∀ τ : L ≃ₐ[K] L,
τ ∈ Subgroup.zpowers σ) :
letI _localAction : ∀ i,
MulDistribMulAction
(absoluteValueDecompositionGroup K (d i).extension.1)
(LocalizedCompletion
(d i).base (d i).extension)ˣ :=
fun i ↦ decompositionGroupLocalUnitsAction
(d i).base (d i).base_isNontrivial
(d i).extension
letI _blockAction : ∀ i,
MulDistribMulAction (L ≃ₐ[K] L)
(LocalPlaceBlock
(d i).base (d i).base_isNontrivial
(d i).extension) :=
fun i ↦ inducedMulDistribMulAction
(absoluteValueDecompositionGroup K (d i).extension.1)
letI _familyAction :
MulDistribMulAction (L ≃ₐ[K] L)
(LocalBlockFamily d) :=
piMulDistribMulAction (L ≃ₐ[K] L)
(fun i ↦ LocalPlaceBlock
(d i).base (d i).base_isNontrivial
(d i).extension)
letI _decompositionFintype : ∀ i,
Fintype
(absoluteValueDecompositionGroup K
(d i).extension.1) :=
fun _ ↦ Fintype.ofFinite _
HerbrandH0 (L ≃ₐ[K] L)
(LocalBlockFamily d) ≃*
∀ i,
HerbrandH0
(absoluteValueDecompositionGroup K
(d i).extension.1)
(LocalizedCompletion
(d i).base (d i).extension)ˣ := by
letI localAction := localBlockFamilyLocalAction d
letI blockAction := localBlockFamilyBlockAction d
letI familyAction := localBlockFamilyCohomologyAction d
letI decompositionFintype :=
localBlockFamilyDecompositionFintype d
exact
(herbrandH0PiEquiv
(G := L ≃ₐ[K] L)
(fun i ↦ LocalPlaceBlock
(d i).base (d i).base_isNontrivial
(d i).extension)).trans
(MulEquiv.piCongrRight fun i ↦
localPlaceBlockHerbrandH0Equiv
(d i).base (d i).base_isNontrivial
(d i).extension σ hgen)