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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
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