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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 | import Mathlib.Algebra.Category.ModuleCat.Ext.HasExt
import Mathlib.Algebra.Homology.DerivedCategory.Ext.ExactSequences
import Mathlib.Data.Nat.Init
import Mathlib.RingTheory.Ideal.AssociatedPrime.Localization
import Mathlib.RingTheory.LocalRing.ResidueField.Basic
import Mathlib.RingTheory.Localization.Submodule
namespace OAI
noncomputable section
universe w v u
namespace PiExponentJets.W18
open CategoryTheory CategoryTheory.Abelian
section ExactDimensionShifting
variable {C : Type u} [Category.{v} C] [Abelian C] [HasExt.{w} C]
theorem ext_subsingleton_of_shortExact
(X : C) (S : ShortComplex C) (hS : S.ShortExact) (n : ℕ)
(hmid : Subsingleton (Ext X S.X₂ n))
(hleft : Subsingleton (Ext X S.X₁ (n + 1))) :
Subsingleton (Ext X S.X₃ n) := by
let := hmid
let := hleft
apply subsingleton_of_forall_eq 0
intro e
obtain ⟨a, ha⟩ := Ext.covariant_sequence_exact₃ X hS e rfl
(Subsingleton.elim _ 0)
rw [Subsingleton.elim a 0, Ext.zero_comp] at ha
exact ha.symm
theorem ext_zero_of_finite_exact_resolution
(X : C) (K F : ℕ → C) (r : ℕ)
(i : ∀ n, K (n + 1) ⟶ F n) (p : ∀ n, F n ⟶ K n)
(hzero : ∀ n, i n ≫ p n = 0)
(hexact : ∀ n < r, (ShortComplex.mk (i n) (p n) (hzero n)).ShortExact)
(hmid : ∀ n < r, Subsingleton (Ext X (F n) n))
(hend : Subsingleton (Ext X (K r) r)) :
Subsingleton (Ext X (K 0) 0) := by
exact Nat.decreasingInduction
(motive := fun n _ => Subsingleton (Ext X (K n) n))
(fun n hn ih => ext_subsingleton_of_shortExact X
(ShortComplex.mk (i n) (p n) (hzero n)) (hexact n hn) n (hmid n hn) ih)
hend (Nat.zero_le r)
theorem nonzero_ext_forces_resolution_obstruction
(X : C) (K F : ℕ → C) (r : ℕ)
(i : ∀ n, K (n + 1) ⟶ F n) (p : ∀ n, F n ⟶ K n)
(hzero : ∀ n, i n ≫ p n = 0)
(hexact : ∀ n < r, (ShortComplex.mk (i n) (p n) (hzero n)).ShortExact)
(e : Ext X (K 0) 0) (he : e ≠ 0) :
¬ ((∀ n < r, Subsingleton (Ext X (F n) n)) ∧
Subsingleton (Ext X (K r) r)) := by
rintro ⟨hmid, hend⟩
let := ext_zero_of_finite_exact_resolution X K F r i p hzero hexact hmid hend
exact he (Subsingleton.elim _ _)
theorem lengthTwo_ext_zero
(X : C) (S T : ShortComplex C) (hS : S.ShortExact) (hT : T.ShortExact)
(hlink : T.X₃ = S.X₁)
(hF₀ : Subsingleton (Ext X S.X₂ 0))
(hF₁ : Subsingleton (Ext X T.X₂ 1))
(hF₂ : Subsingleton (Ext X T.X₁ 2)) :
Subsingleton (Ext X S.X₃ 0) := by
have hK : Subsingleton (Ext X T.X₃ 1) :=
ext_subsingleton_of_shortExact X T hT 1 hF₁ hF₂
rw [hlink] at hK
exact ext_subsingleton_of_shortExact X S hS 0 hF₀ hK
end ExactDimensionShifting
variable {R : Type u} [CommRing R] [IsNoetherianRing R]
variable {M : Type u} [AddCommGroup M] [Module R M]
theorem associatedPrime_nonzero_quotient_hom (P : Ideal R)
(hP : IsAssociatedPrime P M) :
∃ f : R ⧸ P →ₗ[R] M, f ≠ 0 := by
obtain ⟨hprime, f, hf⟩ :=
(isAssociatedPrime_iff_exists_injective_linearMap P M).mp hP
let := hprime
refine ⟨f, ?_⟩
intro hz
have h10 : (1 : R ⧸ P) = 0 := hf (by simp [hz])
exact one_ne_zero h10
theorem associatedPrime_nonzero_ext_zero (P : Ideal R)
(hP : IsAssociatedPrime P M) :
∃ e : Ext (ModuleCat.of R (R ⧸ P)) (ModuleCat.of R M) 0, e ≠ 0 := by
obtain ⟨f, hf⟩ := associatedPrime_nonzero_quotient_hom P hP
refine ⟨Ext.mk₀ (ModuleCat.ofHom f), ?_⟩
intro he
have hh := (Ext.mk₀_eq_zero_iff (ModuleCat.ofHom f)).mp he
exact hf (congrArg (fun g : ModuleCat.of R (R ⧸ P) ⟶ ModuleCat.of R M => g.hom) hh)
theorem not_isAssociatedPrime_of_ext_zero (P : Ideal R)
(h : Subsingleton (Ext (ModuleCat.of R (R ⧸ P)) (ModuleCat.of R M) 0)) :
¬ IsAssociatedPrime P M := by
intro hP
obtain ⟨e, he⟩ := associatedPrime_nonzero_ext_zero P hP
exact he (h.elim e 0)
theorem associatedPrime_local_nonzero_ext_zero (P : Ideal R) [P.IsPrime]
(hP : IsAssociatedPrime P M) :
∃ e : Ext
(ModuleCat.of (Localization.AtPrime P)
(IsLocalRing.ResidueField (Localization.AtPrime P)))
(ModuleCat.of (Localization.AtPrime P) (LocalizedModule.AtPrime P M)) 0,
e ≠ 0 := by
have hlocal : IsAssociatedPrime (IsLocalRing.maximalIdeal (Localization.AtPrime P))
(LocalizedModule.AtPrime P M) :=
Module.associatedPrimes.mem_associatedPrimes_atPrime_of_mem_associatedPrimes hP
exact associatedPrime_nonzero_ext_zero
(IsLocalRing.maximalIdeal (Localization.AtPrime P)) hlocal
end PiExponentJets.W18
end
end OAI
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