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