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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 | import Mathlib.RingTheory.Ideal.KrullsHeightTheorem
import OAI.NumberTheory.PiExponent.Geometry.ConePrimeProperties
import OAI.NumberTheory.PiExponent.LocalAlgebra.ConeRegularSequence
import OAI.NumberTheory.PiExponent.LocalAlgebra.LocalLength
import OAI.NumberTheory.PiExponent.LocalAlgebra.LocalizedRadical
import OAI.NumberTheory.PiExponent.Polynomials.HomogeneousListHistory
namespace OAI
namespace PiExponentJets.W22
open PiExponentJets.W64
open PiExponentSiegel.W17.ConeLocalLength
open scoped BigOperators Classical
theorem span_range_list_get {R : Type*} [CommSemiring R] (gs : List R) :
Ideal.span (Set.range gs.get) = Ideal.ofList gs := by
congr 1
ext x
exact List.mem_iff_get.symm
theorem span_range_map_list_get {R S : Type*} [CommSemiring S]
(gs : List R) (f : R → S) :
Ideal.span (Set.range (fun i => f (gs.get i))) = Ideal.ofList (gs.map f) := by
congr 1
ext x
change (∃ i, f (gs.get i) = x) ↔ x ∈ gs.map f
constructor
· rintro ⟨i,rfl⟩
exact List.mem_map.mpr ⟨gs.get i,List.mem_iff_get.mpr ⟨i,rfl⟩,rfl⟩
· intro hx
obtain ⟨a,ha,rfl⟩ := List.mem_map.mp hx
obtain ⟨i,rfl⟩ := List.mem_iff_get.mp ha
exact ⟨i,rfl⟩
theorem finiteLength_localized_minimal_quotient
{R : Type*} [CommRing R] [IsNoetherianRing R]
(I P : Ideal R) [P.IsPrime] (hP : P ∈ I.minimalPrimes) :
IsFiniteLength (Localization.AtPrime P)
(Localization.AtPrime P ⧸ I.map (algebraMap R (Localization.AtPrime P))) := by
let L := Localization.AtPrime P
let J := I.map (algebraMap R L)
have hrad : J.radical = IsLocalRing.maximalIdeal L :=
localized_radical_eq_maximal_of_mem_minimalPrimes I P hP
let : J.radical.IsPrime := by rw [hrad]; infer_instance
have hmin : IsLocalRing.maximalIdeal L ∈ J.minimalPrimes := by
rw [← hrad]
exact radical_mem_minimalPrimes J
let : IsArtinianRing (L ⧸ J) :=
IsLocalRing.quotient_artinian_of_mem_minimalPrimes_of_isLocalRing J hmin
apply Module.length_ne_top_iff.mp
rw [Module.length_eq_of_surjective (M := L ⧸ J)
(show Function.Surjective (algebraMap L (L ⧸ J)) from Ideal.Quotient.mk_surjective)]
exact Module.length_ne_top
theorem affine_local_regular_list_bezout
{k : Type} [Field k] (n : ℕ)
(P : Ideal (MvPolynomial (Fin n) k)) [P.IsPrime]
(gs : List (MvPolynomial (Fin n) k)) (D : ℕ)
(hdegree : ∀ g ∈ gs, g.totalDegree ≤ D)
(hreg : RingTheory.Sequence.IsRegular (Localization.AtPrime P)
(gs.map (algebraMap (MvPolynomial (Fin n) k) (Localization.AtPrime P))))
(hlen : gs.length ≤ n) (hP : P ∈ (Ideal.ofList gs).minimalPrimes) :
Module.length (Localization.AtPrime P)
(Localization.AtPrime P ⧸ (Ideal.ofList gs).map
(algebraMap (MvPolynomial (Fin n) k) (Localization.AtPrime P))) ≤
((D ^ gs.length : ℕ) : ℕ∞) := by
let hs := gs.map homogenize
let degrees : Fin hs.length → ℕ := fun i =>
(gs[i.val]'(by simpa only [hs,List.length_map] using i.isLt)).totalDegree
have hhom : ∀ i : Fin hs.length, hs[i].IsHomogeneous (degrees i) := by
intro i
simpa only [hs, List.getElem_map, Fin.getElem_fin, degrees] using
homogenize_isHomogeneous (gs[i.val]'(by simpa only [hs,List.length_map] using i.isLt))
have hconeReg := homogenized_list_regular_at_cone k (Fin n) P gs hreg
have hget : P ∈ (Ideal.span (Set.range gs.get)).minimalPrimes := by
simpa only [span_range_list_get] using hP
have hconeMin := conePrime_minimal_over_homogenized k (Fin n) P gs.get hget
rw [span_range_map_list_get] at hconeMin
have hconeLen : hs.length < Fintype.card (Option (Fin n)) := by
simp only [hs,List.length_map,Fintype.card_option,Fintype.card_fin]
omega
have hbound := homogeneous_local_regular_bezout (conePrime k (Fin n) P) none
(homogenizing_coordinate_not_mem_conePrime k (Fin n) P)
hs degrees hhom hconeReg hconeLen hconeMin
have hfinite := finiteLength_localized_minimal_quotient
(Ideal.span (Set.range gs.get)) P hget
have hlength := homogenized_cone_local_length_eq k (Fin n) P gs.get hfinite
rw [span_range_list_get, span_range_map_list_get] at hlength
have hprod : (∏ i ∈ Finset.range hs.length, degreeAt degrees i) ≤ D ^ gs.length := by
calc
_ ≤ ∏ _i ∈ Finset.range hs.length, D := by
apply Finset.prod_le_prod
intro i hi
have hi' : i < hs.length := Finset.mem_range.mp hi
simp only [degreeAt,dite_eq_left hi',degrees]
exact hdegree _ (List.getElem_mem _)
_ = D ^ gs.length := by simp only [Finset.prod_const,Finset.card_range,hs,List.length_map]
calc
_ = Module.length (Localization.AtPrime (conePrime k (Fin n) P))
(Localization.AtPrime (conePrime k (Fin n) P) ⧸ (Ideal.ofList hs).map
(algebraMap (MvPolynomial (Option (Fin n)) k)
(Localization.AtPrime (conePrime k (Fin n) P)))) := hlength.symm
_ ≤ ((∏ i ∈ Finset.range hs.length, degreeAt degrees i : ℕ) : ℕ∞) := hbound
_ ≤ _ := by exact_mod_cast hprod
end PiExponentJets.W22
end OAI
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