openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /LocalAlgebra /AffineLocalRegularBezout.lean
| 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 | |