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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 | 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
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