| 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 |
|
|
