Project-declaredLean 4.32.0 · mathlib@249c48c2

Number Field Finite Adele Ring tensor is Central Simple add Haar Scalar Factor left mul eq right mul

NumberField.FiniteAdeleRing.tensor_isCentralSimple_addHaarScalarFactor_left_mul_eq_right_mul

number theory arithmetic geometry Fermat's Last Theorem finite adele ring

Plain-language statement

left multiplication and right multiplication by a unit have the same Haar character on 𝔸_K^f ⊗ B. See also NumberField.FiniteAdeleRing.isCentralSimple_addHaarScalarFactor_left_mul_eq_right_mul which proves it for B ⊗ 𝔸_K^f.

Exact Lean statement

lemma NumberField.FiniteAdeleRing.tensor_isCentralSimple_addHaarScalarFactor_left_mul_eq_right_mul
    [IsSimpleRing B] [Algebra.IsCentral K B] (u : ((FiniteAdeleRing (𝓞 K) K) ⊗[K] B)ˣ) :
    addEquivAddHaarChar (ContinuousAddEquiv.mulLeft u) =
    addEquivAddHaarChar (ContinuousAddEquiv.mulRight u)

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

lemma NumberField.FiniteAdeleRing.tensor_isCentralSimple_addHaarScalarFactor_left_mul_eq_right_mul    [IsSimpleRing B] [Algebra.IsCentral K B] (u : ((FiniteAdeleRing (𝓞 K) K) ⊗[K] B)ˣ) :    addEquivAddHaarChar (ContinuousAddEquiv.mulLeft u) =    addEquivAddHaarChar (ContinuousAddEquiv.mulRight u) := by  change addEquivAddHaarChar      (ContinuousLinearEquiv.mulLeft ((FiniteAdeleRing (𝓞 K) K)) u).toContinuousAddEquiv =    addEquivAddHaarChar      (ContinuousLinearEquiv.mulRight ((FiniteAdeleRing (𝓞 K) K)) u).toContinuousAddEquiv  rw [FiniteAdeleRing.Aux.f_commSq, FiniteAdeleRing.Aux.f_commSq]  rw [FiniteAdeleRing.Aux.g_commSq, FiniteAdeleRing.Aux.g_commSq]  rw [FiniteAdeleRing.Aux.f_g_local_global, FiniteAdeleRing.Aux.f_g_local_global]  rw [addEquivAddHaarChar_restrictedProductCongrRight,    addEquivAddHaarChar_restrictedProductCongrRight]  congr  ext v  rw [← FiniteAdeleRing.Aux.e_commSq, ← FiniteAdeleRing.Aux.e_commSq]  rw [localcomponent_mulLeft, localcomponent_mulRight]  congr 1  let w : (adicCompletion K v ⊗[K] B)ˣ := ((Units.map (Algebra.TensorProduct.rTensor B    (FiniteAdeleRing.evalContinuousAlgebraMap (𝓞 K) K v).toAlgHom).toMonoidHom) u)  exact IsSimpleRing.ringHaarChar_eq_addEquivAddHaarChar_mulRight (F := v.adicCompletion K) w

Project: Fermat's Last Theorem
License: Apache-2.0
Commit: 8dd808888295
Source: FLT/HaarMeasure/HaarChar/FiniteAdeleRing.lean:622-642

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