| import Mathlib.Algebra.Homology.DerivedCategory.KProjective |
| import Mathlib.Algebra.Homology.HomotopyCategory.ShortExact |
| import OAI.Algebra.Finitistic.Rectification.RectificationTotal |
|
|
| namespace OAI |
|
|
| namespace LittleFinitistic.RectificationAcyclic |
| open CategoryTheory CategoryTheory.Limits HomologicalComplex CochainComplex |
|
|
| variable {C : Type*} [Category C] [Abelian C] |
|
|
| /-- The ordinary cone of a quasi-isomorphism is acyclic. -/ |
| theorem cone_acyclic {K L : CochainComplex C ℤ} (f : K ⟶ L) [QuasiIso f] : |
| (mappingCone f).Acyclic := by |
| let P := HomotopyCategory.subcategoryAcyclic C |
| have hf : P.trW ((HomotopyCategory.quotient C (.up ℤ)).map f) := by |
| rw [← HomotopyCategory.quasiIso_eq_trW_subcategoryAcyclic, |
| HomotopyCategory.quotient_map_mem_quasiIso_iff,HomologicalComplex.mem_quasiIso_iff] |
| infer_instance |
| have hT := HomotopyCategory.mappingCone_triangleh_distinguished f |
| have hz := (P.trW_iff_of_distinguished (mappingCone.triangleh f) hT).1 hf |
| exact (HomotopyCategory.quotient_obj_mem_subcategoryAcyclic_iff_acyclic _).1 hz |
|
|
| /-- The total of a strict short exact sequence of complexes is acyclic. This is |
| applied to the associated graded source-index pieces in rectification. -/ |
| theorem short_exact_total_acyclic (S : ShortComplex (CochainComplex C ℤ)) |
| (hS : S.ShortExact) : |
| (RectificationTotal.total S.g S.f (Homotopy.ofEq S.zero)).Acyclic := by |
| have he : RectificationTotal.rectifyingMap S.g S.f (Homotopy.ofEq S.zero) = |
| mappingCone.descShortComplex S := by |
| simp only [RectificationTotal.rectifyingMap,CochainComplex.HomComplex.Cochain.ofHomotopy_ofEq, |
| mappingCone.descShortComplex] |
| change (mappingCone (RectificationTotal.rectifyingMap S.g S.f (Homotopy.ofEq S.zero))).Acyclic |
| rw [he] |
| let := mappingCone.quasiIso_descShortComplex hS |
| exact cone_acyclic _ |
|
|
| /-- Bounded-above acyclic complexes of projectives admit an actual contraction. -/ |
| noncomputable def contractBoundedProjective (K : CochainComplex C ℤ) (hi : ℤ) |
| [K.IsStrictlyLE hi] [∀ i, Projective (K.X i)] (hK : K.Acyclic) : |
| Homotopy (𝟙 K) 0 := by |
| letI := CochainComplex.isKProjective_of_projective K hi |
| exact CochainComplex.IsKProjective.homotopyZero (𝟙 K) hK |
|
|
| /-- A quasi-isomorphism between bounded-above projective complexes has an actual |
| chain-map inverse up to specified homotopies, retaining the given forward map. -/ |
| theorem bounded_projective_homotopy_inverse {K L : CochainComplex C ℤ} |
| (f : K ⟶ L) [QuasiIso f] (hi : ℤ) [K.IsStrictlyLE hi] [L.IsStrictlyLE hi] |
| [∀ i, Projective (K.X i)] [∀ i, Projective (L.X i)] : |
| ∃ g : L ⟶ K, Nonempty (Homotopy (f ≫ g) (𝟙 K)) ∧ |
| Nonempty (Homotopy (g ≫ f) (𝟙 L)) := by |
| classical |
| have : (mappingCone f).IsStrictlyLE hi := by |
| rw [isStrictlyLE_iff] |
| intro i hi' |
| rw [mappingCone.isZero_X_iff] |
| exact ⟨K.isZero_of_isStrictlyLE hi _ (by omega),L.isZero_of_isStrictlyLE hi _ hi'⟩ |
| have (degree : ℤ) : Projective ((mappingCone f).X degree) := by |
| apply Projective.of_iso (homotopyCofiber.XIsoBiprod f degree (degree+1) (by simp)).symm |
| infer_instance |
| let q := HomotopyCategory.quotient C (.up ℤ) |
| have hzero : IsZero (q.obj (mappingCone f)) := |
| (HomotopyCategory.isZero_quotient_obj_iff _).2 |
| ⟨contractBoundedProjective _ hi (cone_acyclic f)⟩ |
| have : IsIso (q.map f) := |
| ((mappingCone.triangleh f).isZero₃_iff_isIso₁ |
| (HomotopyCategory.mappingCone_triangleh_distinguished f)).1 hzero |
| let g := q.preimage (inv (q.map f)) |
| have hg : q.map g = inv (q.map f) := q.map_preimage _ |
| refine ⟨g,⟨HomotopyCategory.homotopyOfEq _ _ ?_⟩, |
| ⟨HomotopyCategory.homotopyOfEq _ _ ?_⟩⟩ |
| · change q.map (f ≫ g) = q.map (𝟙 K) |
| rw [Functor.map_comp,hg,IsIso.hom_inv_id,q.map_id] |
| · change q.map (g ≫ f) = q.map (𝟙 L) |
| rw [Functor.map_comp,hg,IsIso.inv_hom_id,q.map_id] |
|
|
| /-- A finite decreasing filtration with acyclic successive quotients is acyclic. |
| The hypotheses are actual short exact sequences of complexes. -/ |
| theorem finite_exact_filtration_acyclic (n : ℕ) |
| (F G : ℕ → CochainComplex C ℤ) |
| (u : ∀ i, F (i+1) ⟶ F i) (v : ∀ i, F i ⟶ G i) |
| (huv : ∀ i, u i ≫ v i = 0) |
| (hseq : ∀ i, i < n → (ShortComplex.mk (u i) (v i) (huv i)).ShortExact) |
| (hG : ∀ i, i < n → (G i).Acyclic) (hn : (F n).Acyclic) : |
| (F 0).Acyclic := by |
| apply Nat.decreasingInduction (motive := fun i _ => (F i).Acyclic) ?_ hn (Nat.zero_le n) |
| intro i hi hnext |
| exact (hseq i hi).acyclic_X₂ hnext (hG i hi) |
|
|
| /-- A short exact inclusion with acyclic quotient is a quasi-isomorphism. -/ |
| theorem quasiIso_of_acyclic_quotient (S : ShortComplex (CochainComplex C ℤ)) |
| (hS : S.ShortExact) (hQ : S.X₃.Acyclic) : QuasiIso S.f := by |
| rw [quasiIso_iff] |
| intro i |
| rw [quasiIsoAt_iff_isIso_homologyMap] |
| have : Mono (HomologicalComplex.homologyMap S.f i) := |
| (hS.homology_exact₁ (i-1) i (by change i-1+1=i; omega)).mono_g |
| ((hQ (i-1)).isZero_homology.eq_of_src _ _) |
| have : Epi (HomologicalComplex.homologyMap S.f i) := |
| (hS.homology_exact₂ i).epi_f ((hQ i).isZero_homology.eq_of_tgt _ _) |
| exact isIso_of_mono_of_epi _ |
|
|
| /-- The given inclusion, rather than an unspecified replacement, has an actual |
| homotopy inverse once its quotient is proved acyclic by the finite filtration. -/ |
| theorem inclusion_homotopy_inverse (S : ShortComplex (CochainComplex C ℤ)) |
| (hS : S.ShortExact) (hQ : S.X₃.Acyclic) (hi : ℤ) |
| [S.X₁.IsStrictlyLE hi] [S.X₂.IsStrictlyLE hi] |
| [∀ i, Projective (S.X₁.X i)] [∀ i, Projective (S.X₂.X i)] : |
| ∃ g : S.X₂ ⟶ S.X₁, Nonempty (Homotopy (S.f ≫ g) (𝟙 S.X₁)) ∧ |
| Nonempty (Homotopy (g ≫ S.f) (𝟙 S.X₂)) := by |
| let := quasiIso_of_acyclic_quotient S hS hQ |
| exact bounded_projective_homotopy_inverse S.f hi |
| end LittleFinitistic.RectificationAcyclic |
|
|
| end OAI |
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|
