openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /LocalAlgebra /AffineClosedPointLocalization.lean
| import Mathlib.RingTheory.FiniteType | |
| import Mathlib.RingTheory.Ideal.Quotient.Basic | |
| import Mathlib.RingTheory.Localization.Ideal | |
| import OAI.NumberTheory.PiExponent.LocalAlgebra.QuotientLocalizedLength | |
| import OAI.NumberTheory.PiExponent.Polynomials.PolynomialPrimeClosedPointEquiv | |
| namespace OAI | |
| noncomputable section | |
| namespace PiExponentSiegel.W23 | |
| open PiExponentSiegelAux.W09 | |
| open PiExponentJets.PolynomialLocalResidueResolution | |
| variable (K α β : Type*) [Field K] | |
| def affineCoefficientAlgHom : | |
| SplitPolynomial K α β →ₐ[K] FractionCoefficientPolynomial K α β := | |
| (polynomialCoefficientLocalizationEquiv K α β).toAlgHom.comp | |
| (IsScalarTower.toAlgHom K (SplitPolynomial K α β) | |
| (PolynomialCoefficientLocalization K α β)) | |
| abbrev affineCoefficientMap : | |
| SplitPolynomial K α β →+* FractionCoefficientPolynomial K α β := | |
| (affineCoefficientAlgHom K α β).toRingHom | |
| @[reducible] def affineCoefficientAlgebra : | |
| Algebra (SplitPolynomial K α β) (FractionCoefficientPolynomial K α β) := | |
| (affineCoefficientMap K α β).toAlgebra | |
| attribute [local instance] affineCoefficientAlgebra | |
| instance affineCoefficient_isLocalization : | |
| IsLocalization (coefficientDenominators K α β) (FractionCoefficientPolynomial K α β) := by | |
| exact (IsLocalization.isLocalization_iff_of_ringEquiv | |
| (coefficientDenominators K α β) | |
| (polynomialCoefficientLocalizationEquiv K α β).toRingEquiv).mp inferInstance | |
| def affineMappedIdeal (P : Ideal (SplitPolynomial K α β)) : | |
| Ideal (FractionCoefficientPolynomial K α β) := | |
| P.map (affineCoefficientMap K α β) | |
| variable (Q : Ideal (SplitPolynomial K α β)) [Q.IsPrime] | |
| variable (hB : IsTranscendenceBasis K (splitResidueBeta K α β Q)) | |
| theorem affineCoefficientMap_comap_closedPoint : | |
| (closedPolynomialIdeal K α β Q hB).comap (affineCoefficientMap K α β) = Q := by | |
| change ((coefficientClosedPoint K α β Q hB).map | |
| (polynomialCoefficientLocalizationEquiv K α β).toRingHom).comap | |
| ((polynomialCoefficientLocalizationEquiv K α β).toRingHom.comp | |
| (algebraMap (SplitPolynomial K α β) (PolynomialCoefficientLocalization K α β))) = Q | |
| rw [← Ideal.comap_comap, ← atPrimeImage_comap] | |
| exact coefficientClosedPoint_comap K α β Q hB | |
| include hB in | |
| theorem affineDenominators_disjoint (P : Ideal (SplitPolynomial K α β)) (hPQ : P ≤ Q) : | |
| Disjoint (coefficientDenominators K α β : Set (SplitPolynomial K α β)) (P : Set _) := by | |
| have hd : Disjoint (coefficientDenominators K α β : Set (SplitPolynomial K α β)) | |
| (Q : Set _) := by | |
| rw [← affineCoefficientMap_comap_closedPoint K α β Q hB] | |
| exact (IsLocalization.disjoint_under_iff (coefficientDenominators K α β) | |
| (FractionCoefficientPolynomial K α β) (closedPolynomialIdeal K α β Q hB)).mpr | |
| (Ideal.IsMaximal.ne_top inferInstance) | |
| exact Set.disjoint_left.mpr fun x hx hxP => Set.disjoint_left.mp hd hx (hPQ hxP) | |
| include hB in | |
| theorem affineMappedIdeal_isPrime (P : Ideal (SplitPolynomial K α β)) [P.IsPrime] | |
| (hPQ : P ≤ Q) : (affineMappedIdeal K α β P).IsPrime := by | |
| exact IsLocalization.isPrime_of_isPrime_disjoint (coefficientDenominators K α β) | |
| (FractionCoefficientPolynomial K α β) P inferInstance | |
| (affineDenominators_disjoint K α β Q hB P hPQ) | |
| theorem affineMappedIdeal_le_closedPoint (P : Ideal (SplitPolynomial K α β)) (hPQ : P ≤ Q) : | |
| affineMappedIdeal K α β P ≤ closedPolynomialIdeal K α β Q hB := by | |
| apply Ideal.map_le_iff_le_comap.mpr | |
| rw [affineCoefficientMap_comap_closedPoint] | |
| exact hPQ | |
| include hB in | |
| theorem affineMappedIdeal_comap (P : Ideal (SplitPolynomial K α β)) [P.IsPrime] | |
| (hPQ : P ≤ Q) : | |
| (affineMappedIdeal K α β P).comap (affineCoefficientMap K α β) = P := | |
| IsLocalization.under_map_of_isPrime_disjoint (coefficientDenominators K α β) | |
| (FractionCoefficientPolynomial K α β) inferInstance | |
| (affineDenominators_disjoint K α β Q hB P hPQ) | |
| theorem originalLocalCoefficientLocalEquiv_algebraMap (a : SplitPolynomial K α β) : | |
| originalLocalCoefficientLocalEquiv K α β Q hB | |
| (algebraMap (SplitPolynomial K α β) (Localization.AtPrime Q) a) = | |
| algebraMap (SplitPolynomial K α β) (CoefficientClosedLocal K α β Q hB) a := by | |
| let := closedLocal_isLocalizationAtOriginal K α β Q hB | |
| exact (IsLocalization.algEquiv Q.primeCompl (Localization.AtPrime Q) | |
| (CoefficientClosedLocal K α β Q hB)).commutes a | |
| theorem closedPointLocalEquiv_algebraMap (a : SplitPolynomial K α β) : | |
| polynomialPrimeClosedPointEquiv K α β Q hB | |
| (algebraMap (SplitPolynomial K α β) (Localization.AtPrime Q) a) = | |
| algebraMap (FractionCoefficientPolynomial K α β) | |
| (Localization.AtPrime (closedPolynomialIdeal K α β Q hB)) | |
| (affineCoefficientMap K α β a) := by | |
| change atPrimeEquivOfRingEquiv | |
| (polynomialCoefficientLocalizationEquiv K α β).toRingEquiv | |
| (coefficientClosedPoint K α β Q hB) | |
| (originalLocalCoefficientLocalEquiv K α β Q hB | |
| (algebraMap (SplitPolynomial K α β) (Localization.AtPrime Q) a)) = _ | |
| exact (congrArg (atPrimeEquivOfRingEquiv | |
| (polynomialCoefficientLocalizationEquiv K α β).toRingEquiv | |
| (coefficientClosedPoint K α β Q hB)) | |
| (originalLocalCoefficientLocalEquiv_algebraMap K α β Q hB a)).trans | |
| (atPrimeEquivOfRingEquiv_algebraMap | |
| (polynomialCoefficientLocalizationEquiv K α β).toRingEquiv | |
| (coefficientClosedPoint K α β Q hB) | |
| (algebraMap (SplitPolynomial K α β) (PolynomialCoefficientLocalization K α β) a)) | |
| theorem closedPointLocalEquiv_map_ideal (P : Ideal (SplitPolynomial K α β)) : | |
| (parameterLocalizedIdeal P Q).map | |
| (polynomialPrimeClosedPointEquiv K α β Q hB).toRingHom = | |
| parameterLocalizedIdeal (affineMappedIdeal K α β P) | |
| (closedPolynomialIdeal K α β Q hB) := by | |
| unfold parameterLocalizedIdeal affineMappedIdeal | |
| rw [Ideal.map_map, Ideal.map_map] | |
| congr 1 | |
| apply RingHom.ext | |
| intro a | |
| exact closedPointLocalEquiv_algebraMap K α β Q hB a | |
| def affineLocalQuotientEquiv (P : Ideal (SplitPolynomial K α β)) : | |
| (Localization.AtPrime Q ⧸ parameterLocalizedIdeal P Q) ≃+* | |
| (Localization.AtPrime (closedPolynomialIdeal K α β Q hB) ⧸ | |
| parameterLocalizedIdeal (affineMappedIdeal K α β P) | |
| (closedPolynomialIdeal K α β Q hB)) := | |
| Ideal.quotientEquiv _ _ (polynomialPrimeClosedPointEquiv K α β Q hB) | |
| (closedPointLocalEquiv_map_ideal K α β Q hB P).symm | |
| def affineClosedQuotientLocalEquiv (P : Ideal (SplitPolynomial K α β)) (hPQ : P ≤ Q) : | |
| Localization.AtPrime (quotientParameterPrime P Q hPQ) ≃+* | |
| Localization.AtPrime | |
| (R := FractionCoefficientPolynomial K α β ⧸ affineMappedIdeal K α β P) | |
| (quotientParameterPrime (affineMappedIdeal K α β P) | |
| (closedPolynomialIdeal K α β Q hB) | |
| (affineMappedIdeal_le_closedPoint K α β Q hB P hPQ)) := | |
| ((localizationQuotientParametersEquiv P Q hPQ).symm.trans | |
| (affineLocalQuotientEquiv K α β Q hB P)).trans | |
| (localizationQuotientParametersEquiv (affineMappedIdeal K α β P) | |
| (closedPolynomialIdeal K α β Q hB) | |
| (affineMappedIdeal_le_closedPoint K α β Q hB P hPQ)) | |
| def affineQuotientMap (P : Ideal (SplitPolynomial K α β)) : | |
| (SplitPolynomial K α β ⧸ P) →+* | |
| (FractionCoefficientPolynomial K α β ⧸ affineMappedIdeal K α β P) := | |
| Ideal.quotientMap _ (affineCoefficientMap K α β) Ideal.le_comap_map | |
| @[reducible] def affineQuotientAlgebra (P : Ideal (SplitPolynomial K α β)) : | |
| Algebra (SplitPolynomial K α β ⧸ P) | |
| (FractionCoefficientPolynomial K α β ⧸ affineMappedIdeal K α β P) := | |
| RingHom.toAlgebra' | |
| (R := SplitPolynomial K α β ⧸ P) | |
| (S := FractionCoefficientPolynomial K α β ⧸ affineMappedIdeal K α β P) | |
| (affineQuotientMap K α β P) (fun x y => | |
| (Ideal.Quotient.commSemiring (affineMappedIdeal K α β P)).mul_comm | |
| (affineQuotientMap K α β P x) y) | |
| attribute [local instance] affineQuotientAlgebra | |
| @[simp] theorem affineQuotientMap_mk (P : Ideal (SplitPolynomial K α β)) | |
| (a : SplitPolynomial K α β) : | |
| affineQuotientMap K α β P (Ideal.Quotient.mk P a) = | |
| Ideal.Quotient.mk _ (affineCoefficientMap K α β a) := rfl | |
| instance affineQuotient_scalarTower (P : Ideal (SplitPolynomial K α β)) : | |
| IsScalarTower K (SplitPolynomial K α β ⧸ P) | |
| (FractionCoefficientPolynomial K α β ⧸ affineMappedIdeal K α β P) := by | |
| apply IsScalarTower.of_algebraMap_eq | |
| (R := K) (S := SplitPolynomial K α β ⧸ P) | |
| (A := FractionCoefficientPolynomial K α β ⧸ affineMappedIdeal K α β P) | |
| intro k | |
| change Ideal.Quotient.mk _ (algebraMap K (FractionCoefficientPolynomial K α β) k) = | |
| affineQuotientMap K α β P (Ideal.Quotient.mk P (algebraMap K (SplitPolynomial K α β) k)) | |
| rw [affineQuotientMap_mk] | |
| exact congrArg (Ideal.Quotient.mk (affineMappedIdeal K α β P)) | |
| ((affineCoefficientAlgHom K α β).commutes k).symm | |
| include hB in | |
| theorem affineQuotientMap_injective (P : Ideal (SplitPolynomial K α β)) [P.IsPrime] | |
| (hPQ : P ≤ Q) : Function.Injective (affineQuotientMap K α β P) := | |
| Ideal.quotientMap_injective' (affineMappedIdeal_comap K α β Q hB P hPQ).le | |
| include hB in | |
| theorem affineQuotient_faithfulSMul (P : Ideal (SplitPolynomial K α β)) [P.IsPrime] | |
| (hPQ : P ≤ Q) : | |
| FaithfulSMul (SplitPolynomial K α β ⧸ P) | |
| (FractionCoefficientPolynomial K α β ⧸ affineMappedIdeal K α β P) := | |
| (faithfulSMul_iff_algebraMap_injective _ _).mpr | |
| (affineQuotientMap_injective K α β Q hB P hPQ) | |
| include hB in | |
| theorem affineQuotient_isDomain (P : Ideal (SplitPolynomial K α β)) [P.IsPrime] | |
| (hPQ : P ≤ Q) : | |
| IsDomain (FractionCoefficientPolynomial K α β ⧸ affineMappedIdeal K α β P) := by | |
| let := affineMappedIdeal_isPrime K α β Q hB P hPQ | |
| infer_instance | |
| theorem affineQuotient_finiteType [Finite α] (P : Ideal (SplitPolynomial K α β)) : | |
| Algebra.FiniteType (SplitCoefficientField K β) | |
| (FractionCoefficientPolynomial K α β ⧸ affineMappedIdeal K α β P) := | |
| Algebra.FiniteType.of_surjective | |
| (Ideal.Quotient.mkₐ (SplitCoefficientField K β) (affineMappedIdeal K α β P)) | |
| Ideal.Quotient.mk_surjective | |
| include hB in | |
| theorem affineQuotient_isLocalization (P : Ideal (SplitPolynomial K α β)) [P.IsPrime] | |
| (hPQ : P ≤ Q) : | |
| IsLocalization ((coefficientDenominators K α β).map (Ideal.Quotient.mk P)) | |
| (FractionCoefficientPolynomial K α β ⧸ affineMappedIdeal K α β P) := by | |
| apply (isLocalization_iff | |
| ((coefficientDenominators K α β).map (Ideal.Quotient.mk P)) | |
| (FractionCoefficientPolynomial K α β ⧸ affineMappedIdeal K α β P)).mpr | |
| refine ⟨?_, ?_, ?_⟩ | |
| · intro s | |
| obtain ⟨a, ha, he⟩ := s.property | |
| change IsUnit (affineQuotientMap K α β P (s : SplitPolynomial K α β ⧸ P)) | |
| rw [← he, affineQuotientMap_mk] | |
| exact (IsLocalization.map_units (FractionCoefficientPolynomial K α β) | |
| (⟨a, ha⟩ : coefficientDenominators K α β)).map | |
| (Ideal.Quotient.mk (affineMappedIdeal K α β P)) | |
| · intro z | |
| obtain ⟨t, rfl⟩ := Ideal.Quotient.mk_surjective z | |
| obtain ⟨⟨a, s⟩, hs⟩ := IsLocalization.surj (coefficientDenominators K α β) t | |
| refine ⟨⟨Ideal.Quotient.mk P a, | |
| ⟨Ideal.Quotient.mk P (s : SplitPolynomial K α β), | |
| Submonoid.mem_map_of_mem (Ideal.Quotient.mk P) s.property⟩⟩, ?_⟩ | |
| change Ideal.Quotient.mk _ t * | |
| affineQuotientMap K α β P (Ideal.Quotient.mk P (s : SplitPolynomial K α β)) = | |
| affineQuotientMap K α β P (Ideal.Quotient.mk P a) | |
| rw [affineQuotientMap_mk, affineQuotientMap_mk, ← map_mul] | |
| exact congrArg (Ideal.Quotient.mk (affineMappedIdeal K α β P)) hs | |
| · intro x y hxy | |
| refine ⟨1, ?_⟩ | |
| simpa only [Submonoid.coe_one, one_mul] using | |
| (affineQuotientMap_injective K α β Q hB P hPQ hxy) | |
| theorem affineClosedQuotientLocalEquiv_mk_algebraMap | |
| (P : Ideal (SplitPolynomial K α β)) (hPQ : P ≤ Q) (a : SplitPolynomial K α β) : | |
| affineClosedQuotientLocalEquiv K α β Q hB P hPQ | |
| (algebraMap (SplitPolynomial K α β ⧸ P) | |
| (Localization.AtPrime (quotientParameterPrime P Q hPQ)) (Ideal.Quotient.mk P a)) = | |
| algebraMap | |
| (self := (inferInstance : Algebra | |
| (FractionCoefficientPolynomial K α β ⧸ affineMappedIdeal K α β P) | |
| (Localization.AtPrime (R := FractionCoefficientPolynomial K α β ⧸ affineMappedIdeal K α β P) | |
| (quotientParameterPrime (affineMappedIdeal K α β P) | |
| (closedPolynomialIdeal K α β Q hB) | |
| (affineMappedIdeal_le_closedPoint K α β Q hB P hPQ))))) | |
| (FractionCoefficientPolynomial K α β ⧸ affineMappedIdeal K α β P) | |
| (Localization.AtPrime | |
| (R := FractionCoefficientPolynomial K α β ⧸ affineMappedIdeal K α β P) | |
| (hp := quotientParameterPrime_isPrime (affineMappedIdeal K α β P) | |
| (closedPolynomialIdeal K α β Q hB) | |
| (affineMappedIdeal_le_closedPoint K α β Q hB P hPQ)) | |
| (quotientParameterPrime (affineMappedIdeal K α β P) | |
| (closedPolynomialIdeal K α β Q hB) | |
| (affineMappedIdeal_le_closedPoint K α β Q hB P hPQ))) | |
| (affineQuotientMap K α β P (Ideal.Quotient.mk P a)) := by | |
| have hleft : (localizationQuotientParametersEquiv P Q hPQ).symm | |
| (algebraMap (SplitPolynomial K α β ⧸ P) | |
| (Localization.AtPrime (quotientParameterPrime P Q hPQ)) (Ideal.Quotient.mk P a)) = | |
| Ideal.Quotient.mk (parameterLocalizedIdeal P Q) | |
| (algebraMap (SplitPolynomial K α β) (Localization.AtPrime Q) a) := by | |
| apply (localizationQuotientParametersEquiv P Q hPQ).injective | |
| rw [RingEquiv.apply_symm_apply] | |
| exact (localizationQuotientParametersEquiv_mk_algebraMap P Q hPQ a).symm | |
| simp only [affineClosedQuotientLocalEquiv, RingEquiv.trans_apply] | |
| rw [hleft] | |
| change localizationQuotientParametersEquiv (affineMappedIdeal K α β P) | |
| (closedPolynomialIdeal K α β Q hB) | |
| (affineMappedIdeal_le_closedPoint K α β Q hB P hPQ) | |
| (Ideal.Quotient.mk _ (polynomialPrimeClosedPointEquiv K α β Q hB | |
| (algebraMap (SplitPolynomial K α β) (Localization.AtPrime Q) a))) = _ | |
| rw [closedPointLocalEquiv_algebraMap, | |
| localizationQuotientParametersEquiv_mk_algebraMap, affineQuotientMap_mk] | |
| end PiExponentSiegel.W23 | |
| end | |
| end OAI | |