File size: 4,114 Bytes
b797a5b | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 | 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
|