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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 | import Mathlib.RingTheory.GradedAlgebra.Radical
import Mathlib.RingTheory.Ideal.GoingDown
import Mathlib.RingTheory.Ideal.MinimalPrime.Localization
import Mathlib.RingTheory.KrullDimension.Polynomial
import Mathlib.RingTheory.Localization.Submodule
import OAI.NumberTheory.PiExponent.Polynomials.ConeHomogeneousEquations
namespace OAI
noncomputable section
namespace PiExponentSiegel.W17.ConeLocalLength
attribute [local instance] MvPolynomial.gradedAlgebra
section General
variable {A B : Type*} [CommRing A] [CommRing B]
theorem mapped_minimalPrime_of_prime_extension
(f : A →+* B) {I P : Ideal A} (hP : P ∈ I.minimalPrimes)
(hprime : (P.map f).IsPrime) (hcontract : (P.map f).comap f = P) :
P.map f ∈ (I.map f).minimalPrimes := by
refine ⟨⟨hprime, Ideal.map_mono hP.1.2⟩, ?_⟩
intro Q hQ hQP
apply Ideal.map_le_iff_le_comap.mpr
apply hP.2 ⟨hQ.1.comap f, Ideal.map_le_iff_le_comap.mp hQ.2⟩
exact (Ideal.comap_mono hQP).trans hcontract.le
theorem minimalPrime_isHomogeneous {k σ : Type*} [CommRing k]
(I Q : Ideal (MvPolynomial σ k))
(hI : I.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k))
(hQ : Q ∈ I.minimalPrimes) :
Q.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k) := by
let := MvPolynomial.gradedAlgebra (σ := σ) (R := k)
let C := Q.homogeneousCore (MvPolynomial.homogeneousSubmodule σ k)
have hIC : I ≤ C.toIdeal := by
rw [← hI.toIdeal_homogeneousCore_eq_self]
exact Ideal.homogeneousCore_mono _ hQ.1.2
have hCQ : C.toIdeal ≤ Q := Ideal.toIdeal_homogeneousCore_le _ _
have hQC : Q ≤ C.toIdeal := hQ.2 ⟨hQ.1.1.homogeneousCore, hIC⟩ hCQ
have heq : C.toIdeal = Q := le_antisymm hCQ hQC
exact heq ▸ C.isHomogeneous
end General
variable (k σ : Type*) [CommRing k]
variable (P : Ideal (AffineRing k σ)) [P.IsPrime]
theorem conePrime_minimal_over_homogenized {ι : Type*} (f : ι → AffineRing k σ)
(hP : P ∈ (Ideal.span (Set.range f)).minimalPrimes) :
conePrime k σ P ∈
(Ideal.span (Set.range (fun i => PiExponentJets.W22.homogenize (f i)))).minimalPrimes := by
let H := Ideal.span (Set.range (fun i => PiExponentJets.W22.homogenize (f i)))
have hLP : laurentPrime (AffineRing k σ) P ∈ (H.map (coneToLaurent k σ)).minimalPrimes := by
rw [map_homogenized_span_eq_laurent_span k σ f]
exact mapped_minimalPrime_of_prime_extension LaurentPolynomial.C hP inferInstance
(laurentPrime_comap (AffineRing k σ) P)
have hcomp : coneToLaurent k σ = (coneLaurentEquiv k σ).toRingHom.comp
(algebraMap (ConeRing k σ) (ConeAway k σ)) := by
apply RingHom.ext
intro F
exact (coneAwayToLaurent_algebraMap k σ F).symm
have hI : (H.map (coneToLaurent k σ)).comap (coneLaurentEquiv k σ).toRingHom =
H.map (algebraMap (ConeRing k σ) (ConeAway k σ)) := by
rw [hcomp, ← Ideal.map_map]
simp only [RingEquiv.toRingHom_eq_coe]
rw [Ideal.map_comap_of_equiv]
apply Ideal.ext
intro z
change (coneLaurentEquiv k σ).symm ((coneLaurentEquiv k σ) z) ∈
H.map (algebraMap (ConeRing k σ) (ConeAway k σ)) ↔ z ∈
H.map (algebraMap (ConeRing k σ) (ConeAway k σ))
rw [RingEquiv.symm_apply_apply]
have hchart := Ideal.minimalPrimes_comap_of_surjective
(coneLaurentEquiv k σ).surjective hLP
change awayChartPrime k σ P ∈
((H.map (coneToLaurent k σ)).comap (coneLaurentEquiv k σ).toRingHom).minimalPrimes at hchart
rw [hI, IsLocalization.minimalPrimes_map
(Submonoid.powers (MvPolynomial.X none : ConeRing k σ)) (ConeAway k σ)] at hchart
exact hchart
theorem homogenized_span_isHomogeneous {ι : Type*} (f : ι → AffineRing k σ) :
(Ideal.span (Set.range (fun i => PiExponentJets.W22.homogenize (f i)))).IsHomogeneous
(MvPolynomial.homogeneousSubmodule (Option σ) k) := by
let := MvPolynomial.gradedAlgebra (σ := Option σ) (R := k)
apply Ideal.homogeneous_span (MvPolynomial.homogeneousSubmodule (Option σ) k)
rintro F ⟨i, rfl⟩
exact ⟨(f i).totalDegree, PiExponentJets.W22.homogenize_isHomogeneous (f i)⟩
theorem conePrime_isHomogeneous_of_minimal {ι : Type*} (f : ι → AffineRing k σ)
(hP : P ∈ (Ideal.span (Set.range f)).minimalPrimes) :
(conePrime k σ P).IsHomogeneous (MvPolynomial.homogeneousSubmodule (Option σ) k) :=
minimalPrime_isHomogeneous _ _ (homogenized_span_isHomogeneous k σ f)
(conePrime_minimal_over_homogenized k σ P f hP)
variable [IsNoetherianRing (AffineRing k σ)]
theorem laurentPrime_height_eq :
(laurentPrime (AffineRing k σ) P).height = P.height := by
let : IsNoetherianRing (LaurentChart k σ) :=
IsLocalization.isNoetherianRing
(Submonoid.powers (Polynomial.X : Polynomial (AffineRing k σ)))
(LaurentChart k σ) inferInstance
have h := Ideal.height_eq_height_add_of_liesOver_of_hasGoingDown
P (laurentPrime (AffineRing k σ) P)
change (laurentPrime (AffineRing k σ) P).height = P.height +
((laurentPrime (AffineRing k σ) P).map
(Ideal.Quotient.mk (laurentPrime (AffineRing k σ) P))).height at h
simpa only [Ideal.map_quotient_self, Ideal.height_bot, add_zero] using h
theorem conePrime_height_eq : (conePrime k σ P).height = P.height := by
change ((awayChartPrime k σ P).comap
(algebraMap (ConeRing k σ) (ConeAway k σ))).height = P.height
rw [IsLocalization.height_under
(Submonoid.powers (MvPolynomial.X none : ConeRing k σ))]
change ((laurentPrime (AffineRing k σ) P).comap (coneLaurentEquiv k σ).toRingHom).height = _
exact ((coneLaurentEquiv k σ).height_comap (laurentPrime (AffineRing k σ) P)).trans
(laurentPrime_height_eq k σ P)
end PiExponentSiegel.W17.ConeLocalLength
end
end OAI
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