openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /Polynomials /PolynomialAssociatedLocalization.lean
| import Mathlib.Algebra.Module.LocalizedModule.IsLocalization | |
| import Mathlib.Algebra.Module.LocalizedModule.Submodule | |
| import Mathlib.RingTheory.Flat.Localization | |
| import Mathlib.RingTheory.Ideal.AssociatedPrime.Localization | |
| import Mathlib.RingTheory.Localization.LocalizationLocalization | |
| import Mathlib.RingTheory.Polynomial.Basic | |
| import Mathlib.RingTheory.Regular.RegularSequence | |
| import OAI.NumberTheory.PiExponent.LocalAlgebra.RegularPairExtObstruction | |
| namespace OAI | |
| noncomputable section | |
| universe u | |
| open IsLocalRing RingTheory.Sequence | |
| namespace PiExponentSiegel.W58 | |
| section Flat | |
| variable {R S : Type*} [CommRing R] [CommRing S] [Algebra R S] | |
| theorem weaklyRegular_map_of_flat [Module.Flat R S] (rs : List R) | |
| (h : IsWeaklyRegular R rs) : | |
| IsWeaklyRegular S (rs.map (algebraMap R S)) := by | |
| apply (isWeaklyRegular_map_algebraMap_iff S S rs).mpr | |
| exact ((TensorProduct.rid R S).isWeaklyRegular_congr rs).mp | |
| (h.isWeaklyRegular_lTensor (M₂ := S)) | |
| theorem regular_map_of_flat_of_mem_maximalIdeal [Module.Flat R S] | |
| [IsLocalRing S] (rs : List R) (h : IsRegular R rs) | |
| (hm : ∀ r ∈ rs, algebraMap R S r ∈ maximalIdeal S) : | |
| IsRegular S (rs.map (algebraMap R S)) := by | |
| apply (IsLocalRing.isRegular_iff_isWeaklyRegular_of_subset_maximalIdeal | |
| (M := S) (rs := rs.map (algebraMap R S)) ?_).mpr | |
| · exact weaklyRegular_map_of_flat rs h.toIsWeaklyRegular | |
| · intro x hx | |
| obtain ⟨r, hr, rfl⟩ := List.mem_map.mp hx | |
| exact hm r hr | |
| end Flat | |
| section LocalQuotient | |
| variable {R : Type*} [CommRing R] | |
| variable (L : Type*) [CommRing L] [Algebra R L] | |
| variable (P : Ideal R) [P.IsPrime] [IsLocalization.AtPrime L P] [IsLocalRing L] | |
| omit [IsLocalRing L] in | |
| theorem localized_ideal_eq_map (I : Ideal R) : | |
| I.localized' L P.primeCompl (Algebra.linearMap R L) = | |
| I.map (algebraMap R L) := by | |
| rw [Submodule.localized'_eq_span] | |
| rfl | |
| def localizedIdealQuotientEquiv (I : Ideal R) : | |
| (L ⧸ I.localized' L P.primeCompl (Algebra.linearMap R L)) ≃ₗ[L] | |
| L ⧸ I.map (algebraMap R L) := | |
| Submodule.quotEquivOfEq _ _ (localized_ideal_eq_map L P I) | |
| theorem associatedPrime_quotient_atPrime (I : Ideal R) | |
| (hP : IsAssociatedPrime P (R ⧸ I)) : | |
| IsAssociatedPrime (maximalIdeal L) (L ⧸ I.map (algebraMap R L)) := by | |
| apply (localizedIdealQuotientEquiv L P I).isAssociatedPrime_iff.mp | |
| apply Module.associatedPrimes.mem_associatedPrimes_of_comap_mem_associatedPrimes_of_isLocalizedModule | |
| P.primeCompl (I.toLocalizedQuotient' L P.primeCompl (Algebra.linearMap R L)) | |
| change IsAssociatedPrime ((maximalIdeal L).under R) (R ⧸ I) | |
| rwa [IsLocalization.AtPrime.under_maximalIdeal L P] | |
| omit [P.IsPrime] in | |
| theorem ideal_le_of_quotient_associatedPrime (I : Ideal R) | |
| (hP : IsAssociatedPrime P (R ⧸ I)) : I ≤ P := by | |
| simpa only [Submodule.annihilator_top, Ideal.annihilator_quotient] using | |
| hP.annihilator_le | |
| theorem regular_list_at_associatedPrime | |
| (rs : List R) (hreg : IsRegular R rs) | |
| (hP : IsAssociatedPrime P (R ⧸ Ideal.ofList rs)) : | |
| IsRegular L (rs.map (algebraMap R L)) := by | |
| let : Module.Flat R L := IsLocalization.flat L P.primeCompl | |
| apply regular_map_of_flat_of_mem_maximalIdeal rs hreg | |
| intro r hr | |
| have hmem := ideal_le_of_quotient_associatedPrime P (Ideal.ofList rs) hP | |
| (Ideal.subset_span hr) | |
| change r ∈ (maximalIdeal L).under R | |
| rwa [IsLocalization.AtPrime.under_maximalIdeal L P] | |
| end LocalQuotient | |
| variable {R : Type u} [CommRing R] | |
| variable (L : Type u) [CommRing L] [Algebra R L] [IsNoetherianRing L] | |
| variable (P : Ideal R) [P.IsPrime] [IsLocalization.AtPrime L P] | |
| variable [IsLocalRing L] | |
| theorem regular_pair_localized_ext_obstruction (f g : R) | |
| (hreg : IsRegular R [f, g]) | |
| (hP : IsAssociatedPrime P (R ⧸ Ideal.ofList [f, g])) : | |
| ∃ i : ℕ, i ≤ 2 ∧ ¬ Subsingleton | |
| (CategoryTheory.Abelian.Ext | |
| (ModuleCat.of L (L ⧸ maximalIdeal L)) (ModuleCat.of L L) i) := by | |
| have hr := regular_list_at_associatedPrime L P [f, g] hreg hP | |
| have ha := associatedPrime_quotient_atPrime L P (Ideal.ofList [f, g]) hP | |
| rw [Ideal.map_ofList] at ha | |
| simp only [List.map_cons, List.map_nil] at hr ha | |
| exact PiExponentJets.W18.regular_pair_associatedPrime_ext_obstruction | |
| (algebraMap R L f) (algebraMap R L g) hr (maximalIdeal L) ha | |
| end PiExponentSiegel.W58 | |
| end | |
| end OAI | |