| import OAI.Algebra.AuslanderReiten.Basic | |
| import OAI.Algebra.AuslanderReiten.Complete.CompleteResolutionTail | |
| import Mathlib.CategoryTheory.Abelian.Projective.Ext | |
| namespace OAI | |
| open CategoryTheory | |
| namespace ArExplicit | |
| universe u | |
| variable (R : Type u) [Ring R] | |
| (M N : Type u) [AddCommGroup M] [Module R M] | |
| [AddCommGroup N] [Module R N] | |
| theorem positiveExtVanish_iff_resolution | |
| (P : ProjectiveResolution (ModuleCat.of R M)) : | |
| PositiveExtVanish R M N ↔ | |
| ∀ (n : ℕ) (f : P.complex.X (n + 1) ⟶ ModuleCat.of R N), | |
| P.complex.d (n + 2) (n + 1) ≫ f = 0 → | |
| ∃ g : P.complex.X n ⟶ ModuleCat.of R N, | |
| P.complex.d (n + 1) n ≫ g = f := by | |
| constructor | |
| · intro h n f hf | |
| have := h (n + 1) (Nat.zero_lt_succ n) | |
| exact (P.extMk_eq_zero_iff f (n + 2) rfl hf n rfl).mp | |
| (Subsingleton.elim _ _) | |
| · intro h n hn | |
| obtain ⟨p, rfl⟩ := Nat.exists_eq_succ_of_ne_zero (Nat.ne_of_gt hn) | |
| have hz : ∀ a : CategoryTheory.Abelian.Ext | |
| (ModuleCat.of R M) (ModuleCat.of R N) (p + 1), a = 0 := by | |
| intro a | |
| obtain ⟨f, hf, rfl⟩ := P.extMk_surjective a (p + 2) rfl | |
| exact (P.extMk_eq_zero_iff f (p + 2) rfl hf p rfl).mpr (h p f hf) | |
| exact ⟨fun a b => (hz a).trans (hz b).symm⟩ | |
| theorem positiveExtVanish_iff_completeResolution | |
| (P : ChainComplex (ModuleCat.{u} R) ℤ) | |
| (hproj : ∀ n : ℤ, Module.Projective R (P.X n)) | |
| (hex : ∀ n : ℤ, LinearMap.range (P.d (n + 1) n).hom = | |
| LinearMap.ker (P.d n (n - 1)).hom) | |
| (e : M ≃ₗ[R] CompleteCokernel P) : | |
| PositiveExtVanish R M N ↔ | |
| ∀ (n : ℕ) (f : P.X ((n + 1 : ℕ) : ℤ) →ₗ[R] N), | |
| f.comp (P.d ((n + 2 : ℕ) : ℤ) ((n + 1 : ℕ) : ℤ)).hom = 0 → | |
| ∃ g : P.X (n : ℤ) →ₗ[R] N, | |
| f = g.comp (P.d ((n + 1 : ℕ) : ℤ) (n : ℤ)).hom := by | |
| let Q := transportProjectiveResolution (completeCokernelResolution P hproj hex) | |
| e.symm.toModuleIso | |
| refine (positiveExtVanish_iff_resolution R M N Q).trans ⟨?_, ?_⟩ | |
| · intro h n f hf | |
| have hfc : Q.complex.d (n + 2) (n + 1) ≫ ModuleCat.ofHom f = 0 := by | |
| change (nonnegativeTail P).d (n + 1 + 1) (n + 1) ≫ ModuleCat.ofHom f = 0 | |
| rw [nonnegativeTail_d] | |
| exact ModuleCat.hom_ext hf | |
| obtain ⟨g, hg⟩ := h n (ModuleCat.ofHom f) hfc | |
| refine ⟨g.hom, ?_⟩ | |
| change (nonnegativeTail P).d (n + 1) n ≫ g = ModuleCat.ofHom f at hg | |
| rw [nonnegativeTail_d] at hg | |
| exact (congrArg ModuleCat.Hom.hom hg).symm | |
| · intro h n f hf | |
| change (nonnegativeTail P).d (n + 1 + 1) (n + 1) ≫ f = 0 at hf | |
| rw [nonnegativeTail_d] at hf | |
| obtain ⟨g, hg⟩ := h n f.hom (congrArg ModuleCat.Hom.hom hf) | |
| refine ⟨ModuleCat.ofHom g, ?_⟩ | |
| change (nonnegativeTail P).d (n + 1) n ≫ ModuleCat.ofHom g = f | |
| rw [nonnegativeTail_d] | |
| exact ModuleCat.hom_ext hg.symm | |
| theorem positiveExtVanish_of_completeResolution | |
| (P : ChainComplex (ModuleCat.{u} R) ℤ) | |
| (hproj : ∀ n : ℤ, Module.Projective R (P.X n)) | |
| (hex : ∀ n : ℤ, LinearMap.range (P.d (n + 1) n).hom = | |
| LinearMap.ker (P.d n (n - 1)).hom) | |
| (hdual : ∀ (n : ℤ) (f : P.X n →ₗ[R] N), | |
| f.comp (P.d (n + 1) n).hom = 0 → | |
| ∃ g : P.X (n - 1) →ₗ[R] N, f = g.comp (P.d n (n - 1)).hom) | |
| (e : M ≃ₗ[R] CompleteCokernel P) : PositiveExtVanish R M N := by | |
| let Q := transportProjectiveResolution (completeCokernelResolution P hproj hex) | |
| e.symm.toModuleIso | |
| apply (positiveExtVanish_iff_resolution R M N Q).2 | |
| intro n f hf | |
| change (nonnegativeTail P).d (n + 1 + 1) (n + 1) ≫ f = 0 at hf | |
| rw [nonnegativeTail_d] at hf | |
| have aux (i j l : ℤ) (hi : i = j + 1) (hl : l = j - 1) | |
| (f : P.X j →ₗ[R] N) (hf : f.comp (P.d i j).hom = 0) : | |
| ∃ g : P.X l →ₗ[R] N, f = g.comp (P.d j l).hom := by | |
| subst i | |
| subst l | |
| exact hdual j f hf | |
| obtain ⟨g, hg⟩ := aux ((n + 1 + 1 : ℕ) : ℤ) ((n + 1 : ℕ) : ℤ) (n : ℤ) | |
| (by omega) (by omega) f.hom (congrArg ModuleCat.Hom.hom hf) | |
| refine ⟨ModuleCat.ofHom g, ?_⟩ | |
| change (nonnegativeTail P).d (n + 1) n ≫ ModuleCat.ofHom g = f | |
| rw [nonnegativeTail_d] | |
| exact ModuleCat.hom_ext hg.symm | |
| theorem GorensteinProjective.positiveExt_regular | |
| (h : GorensteinProjective R M) : PositiveExtVanish R M R := by | |
| obtain ⟨P, hp, he, hd, ⟨e⟩⟩ := h | |
| exact positiveExtVanish_of_completeResolution R M R P (fun n => (hp n).2) he hd e | |
| end ArExplicit | |
| end OAI | |