| import Mathlib.RingTheory.LocalRing.RingHom.Basic |
| import OAI.NumberTheory.PiExponent.LocalAlgebra.ConeLocalLength |
| import OAI.NumberTheory.PiExponent.LocalAlgebra.LocalizedLengthEquivalence |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace PiExponentSiegel.W17.ConeLocalLength |
|
|
| attribute [local instance] Localization.AtPrime.algebraOfLiesOver |
|
|
| variable (k σ : Type*) [CommRing k] |
| variable (P : Ideal (AffineRing k σ)) [P.IsPrime] |
|
|
| def awayChartPrime : Ideal (ConeAway k σ) := |
| (laurentPrime (AffineRing k σ) P).comap (coneLaurentEquiv k σ).toRingHom |
| |
| instance awayChartPrime_isPrime : (awayChartPrime k σ P).IsPrime := by |
| unfold awayChartPrime |
| infer_instance |
| |
| def conePrime : Ideal (ConeRing k σ) := |
| (awayChartPrime k σ P).comap (algebraMap (ConeRing k σ) (ConeAway k σ)) |
| |
| instance conePrime_isPrime : (conePrime k σ P).IsPrime := by |
| unfold conePrime |
| infer_instance |
| |
| theorem homogenizing_coordinate_not_mem_conePrime : |
| (MvPolynomial.X none : ConeRing k σ) ∉ conePrime k σ P := by |
| intro h |
| change algebraMap (ConeRing k σ) (ConeAway k σ) (MvPolynomial.X none) ∈ |
| awayChartPrime k σ P at h |
| exact (show (awayChartPrime k σ P).IsPrime from inferInstance).ne_top |
| ((awayChartPrime k σ P).eq_top_of_isUnit_mem h |
| (IsLocalization.Away.algebraMap_isUnit (S := ConeAway k σ) |
| (MvPolynomial.X none : ConeRing k σ))) |
| |
| def conePrimeLocalizationEquiv : |
| Localization.AtPrime (conePrime k σ P) ≃+* |
| Localization.AtPrime (laurentPrime (AffineRing k σ) P) := |
| (IsLocalization.localizationLocalizationAtPrimeIsoLocalization |
| (Submonoid.powers (MvPolynomial.X none : ConeRing k σ)) |
| (awayChartPrime k σ P)).toRingEquiv.trans |
| (PiExponentJets.W22.primeLocalizationEquiv (coneLaurentEquiv k σ) |
| (awayChartPrime k σ P) (laurentPrime (AffineRing k σ) P) rfl) |
| |
| def affineLocalToConeLocal : Localization.AtPrime P →+* |
| Localization.AtPrime (conePrime k σ P) := |
| (conePrimeLocalizationEquiv k σ P).symm.toRingHom.comp |
| (algebraMap (Localization.AtPrime P) |
| (Localization.AtPrime (laurentPrime (AffineRing k σ) P))) |
| |
| @[instance_reducible] def affineLocalConeAlgebra : |
| Algebra (Localization.AtPrime P) (Localization.AtPrime (conePrime k σ P)) := |
| (affineLocalToConeLocal k σ P).toAlgebra |
| |
| theorem affineLocalToConeLocal_map_maximalIdeal : |
| (IsLocalRing.maximalIdeal (Localization.AtPrime P)).map |
| (affineLocalToConeLocal k σ P) = |
| IsLocalRing.maximalIdeal (Localization.AtPrime (conePrime k σ P)) := by |
| rw [affineLocalToConeLocal, ← Ideal.map_map, |
| laurentLocal_maximalIdeal_map_eq (AffineRing k σ) P] |
| exact IsLocalRing.map_maximalIdeal_of_surjective |
| (conePrimeLocalizationEquiv k σ P).symm.toRingHom |
| (conePrimeLocalizationEquiv k σ P).symm.surjective |
| |
| theorem affineLocalToConeLocal_flat : |
| letI := affineLocalConeAlgebra k σ P |
| Module.Flat (Localization.AtPrime P) (Localization.AtPrime (conePrime k σ P)) := by |
| let := affineLocalConeAlgebra k σ P |
| let e : Localization.AtPrime (conePrime k σ P) ≃ₐ[Localization.AtPrime P] |
| Localization.AtPrime (laurentPrime (AffineRing k σ) P) := |
| { conePrimeLocalizationEquiv k σ P with |
| commutes' := by |
| intro a |
| change conePrimeLocalizationEquiv k σ P |
| ((conePrimeLocalizationEquiv k σ P).symm (algebraMap _ _ a)) = _ |
| exact (conePrimeLocalizationEquiv k σ P).apply_symm_apply _ } |
| exact Module.Flat.of_linearEquiv e.toLinearEquiv |
| |
| theorem coneLocal_quotient_length |
| (I : Ideal (Localization.AtPrime P)) |
| (hI : IsFiniteLength (Localization.AtPrime P) (Localization.AtPrime P ⧸ I)) : |
| Module.length (Localization.AtPrime (conePrime k σ P)) |
| (Localization.AtPrime (conePrime k σ P) ⧸ I.map (affineLocalToConeLocal k σ P)) = |
| Module.length (Localization.AtPrime P) (Localization.AtPrime P ⧸ I) := by |
| let := affineLocalConeAlgebra k σ P |
| let := affineLocalToConeLocal_flat k σ P |
| exact PiExponentJets.W25.length_quotient_map_of_flat_local |
| (affineLocalToConeLocal_map_maximalIdeal k σ P) I hI |
| |
| end PiExponentSiegel.W17.ConeLocalLength |
| |
| end |
| |
| end OAI |
| |
