| import Mathlib |
| import OAI.Algebra.FinitisticAsymmetry.Triangular.Augmentation |
| import OAI.Algebra.FinitisticAsymmetry.Derived.UpperTruncation |
| import OAI.Algebra.FinitisticAsymmetry.Derived.ExactExtension |
| import OAI.Algebra.FinitisticAsymmetry.Derived.ModuleGeneration |
| import OAI.Algebra.FinitisticAsymmetry.Triangular.InjectiveResolution |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open CategoryTheory CategoryTheory.Limits CategoryTheory.Pretriangulated HomologicalComplex |
| namespace LittleFinitistic.CummingsConstruction |
| universe u w |
| variable {k C ι : Type u} [Field k] [Ring C] [Algebra k C] |
| variable (χ : C →ₐ[k] (ι → k)) (i : ι) |
|
|
| lemma periodicOld_d (m n : ℕ) : (periodicOld χ i).d m n = 0 := by |
| by_cases h : m = n+1 |
| · subst m |
| simp [periodicOld] |
| rfl |
| · exact (periodicOld χ i).shape m n (by simpa [ComplexShape.down_Rel, eq_comm] using h) |
|
|
| def oldSingleι (n : ℕ) : (HomologicalComplex.single (ModuleCat (T χ)) (.down ℕ) n).obj |
| (oldSimple χ i) ⟶ periodicOld χ i := |
| mkHomFromSingle (𝟙 _) (fun j hj => by simp [periodicOld_d]; exact Category.id_comp _) |
| |
| def oldSingleπ (n : ℕ) : periodicOld χ i ⟶ |
| (HomologicalComplex.single (ModuleCat (T χ)) (.down ℕ) n).obj (oldSimple χ i) := |
| mkHomToSingle (𝟙 _) (fun j hj => by simp [periodicOld_d]; exact Category.comp_id _) |
| |
| lemma oldSingleι_π (n : ℕ) : oldSingleι χ i n ≫ oldSingleπ χ i n = 𝟙 _ := by |
| apply from_single_hom_ext |
| simp [oldSingleι, oldSingleπ] |
| erw [mkHomFromSingle_f, mkHomToSingle_f, Category.comp_id, Category.id_comp, |
| Iso.hom_inv_id] |
|
|
| abbrev extendedInjective := (periodicInjective χ i).extend ComplexShape.embeddingDownNat |
| abbrev extendedRegular := (periodicRegular χ i).extend ComplexShape.embeddingDownNat |
| abbrev extendedOld := (periodicOld χ i).extend ComplexShape.embeddingDownNat |
|
|
| variable [HasDerivedCategory.{w} (ModuleCat.{u} (T χ))] |
|
|
| def regularDerivedIso : DerivedCategory.Q.obj (extendedRegular χ i) ≅ |
| (DerivedCategory.singleFunctor (ModuleCat (T χ)) 0).obj (newSimple χ i) := |
| asIso (DerivedCategory.Q.map (extendMap (periodicAug χ i) ComplexShape.embeddingDownNat)) ≪≫ |
| DerivedCategory.Q.mapIso (extendSingleIso ComplexShape.embeddingDownNat (newSimple χ i) 0 0 rfl) ≪≫ |
| ((DerivedCategory.singleFunctorIsoCompQ (ModuleCat (T χ)) 0).app (newSimple χ i)).symm |
|
|
| def oldSingleDerivedIso (n : ℕ) : |
| DerivedCategory.Q.obj (((HomologicalComplex.single (ModuleCat (T χ)) (.down ℕ) n).obj |
| (oldSimple χ i)).extend ComplexShape.embeddingDownNat) ≅ |
| (DerivedCategory.singleFunctor (ModuleCat (T χ)) (-(n : ℤ))).obj (oldSimple χ i) := |
| DerivedCategory.Q.mapIso (extendSingleIso ComplexShape.embeddingDownNat (oldSimple χ i) n (-(n : ℤ)) rfl) ≪≫ |
| ((DerivedCategory.singleFunctorIsoCompQ (ModuleCat (T χ)) (-n)).app (oldSimple χ i)).symm |
|
|
| omit [HasDerivedCategory (ModuleCat (T χ))] in |
| lemma extended_shortExact : |
| ((ShortComplex.mk (periodicι χ i) (periodicπ χ i) (periodicι_π χ i)).map |
| (ComplexShape.embeddingDownNat.extendFunctor (ModuleCat (T χ)))).ShortExact := |
| DerivedSupport.extend_shortExact _ _ (periodic_shortExact χ i) |
|
|
| variable (P : ObjectProperty (DerivedCategory (ModuleCat.{u} (T χ)))) |
| [P.IsClosedUnderIsomorphisms] [P.IsTriangulated] |
| [P.IsClosedUnderColimitsOfShape (Discrete ℕ)] |
| (hI : ∀ M : ModuleCat.{u} (T χ), Injective M → |
| P ((DerivedCategory.singleFunctor (ModuleCat (T χ)) 0).obj M)) |
|
|
| include hI |
|
|
| lemma extendedInjective_mem [DecidableEq ι] : P (DerivedCategory.Q.obj (extendedInjective χ i)) := by |
| apply DerivedSupport.prop_of_boundedAbove_terms P _ 0 |
| · intro j hj |
| exact CochainComplex.isZero_of_isStrictlyLE _ 0 j hj |
| · intro j |
| apply hI |
| by_cases hj : ∃ n : ℕ, ComplexShape.embeddingDownNat.f n = j |
| · obtain ⟨n, hn⟩ := hj |
| exact Injective.of_iso ((periodicInjective χ i).extendXIso _ hn).symm |
| (newInjective_injective χ i) |
| · exact (isZero_extend_X _ _ j (fun n hn => hj ⟨n,hn⟩)).injective |
|
|
| lemma oldSimple_shifted_retract [DecidableEq ι] : |
| P ((DerivedCategory.singleFunctor (ModuleCat (T χ)) (-2)).obj (oldSimple χ i)) := by |
| let S := (ShortComplex.mk (periodicι χ i) (periodicπ χ i) (periodicι_π χ i)).map |
| (ComplexShape.embeddingDownNat.extendFunctor (ModuleCat (T χ))) |
| let hS := extended_shortExact χ i |
| let Δ := DerivedCategory.triangleOfSES hS |
| have hΔ : Δ ∈ distTriang _ := DerivedCategory.triangleOfSES_distinguished hS |
| let F := ComplexShape.embeddingDownNat.extendFunctor (ModuleCat (T χ)) ⋙ DerivedCategory.Q |
| let j := F.map (oldSingleι χ i 2) |
| let p := F.map (oldSingleπ χ i 2) |
| have hjp : j ≫ p = 𝟙 _ := by |
| dsimp [j,p] |
| rw [← F.map_comp, oldSingleι_π, F.map_id] |
| have hzero : j ≫ Δ.mor₃ = 0 := by |
| have hle : (DerivedCategory.TStructure.t (C := ModuleCat (T χ))).IsLE (F.obj _) (-2) := |
| DerivedCategory.TStructure.t.isLE_of_iso (oldSingleDerivedIso χ i 2).symm (-2) |
| have hge : (DerivedCategory.TStructure.t (C := ModuleCat (T χ))).IsGE Δ.obj₁ 0 := |
| DerivedCategory.TStructure.t.isGE_of_iso (regularDerivedIso χ i).symm 0 |
| have hges := DerivedCategory.TStructure.t.isGE_shift Δ.obj₁ 0 1 (-1) (by omega) |
| exact DerivedCategory.TStructure.t.zero (j ≫ Δ.mor₃) (-2) (-1) (by omega) |
| obtain ⟨l, hl⟩ := Triangle.coyoneda_exact₃ Δ hΔ j hzero |
| have hlp : l ≫ (Δ.mor₂ ≫ p) = 𝟙 _ := by erw [← Category.assoc, ← hl, hjp] |
| apply P.prop_of_iso (oldSingleDerivedIso χ i 2) |
| exact DerivedSupport.localizing_of_retract l (Δ.mor₂ ≫ p) hlp P |
| (extendedInjective_mem χ i P hI) |
|
|
| lemma oldSimple_mem [DecidableEq ι] : |
| P ((DerivedCategory.singleFunctor (ModuleCat (T χ)) 0).obj (oldSimple χ i)) := by |
| apply P.prop_of_iso (((DerivedCategory.singleFunctors (ModuleCat (T χ))).shiftIso |
| (-2) 0 (-2) (by omega)).app (oldSimple χ i)) |
| exact P.le_shift (-2) _ (oldSimple_shifted_retract χ i P hI) |
|
|
| lemma newRegular_mem [DecidableEq ι] : |
| P ((DerivedCategory.singleFunctor (ModuleCat (T χ)) 0).obj (newRegular χ i)) := by |
| exact DerivedSupport.prop_single_shortExact₁ P _ (newRegular_exact χ i) |
| (hI _ (newInjective_injective χ i)) (oldSimple_mem χ i P hI) |
|
|
| lemma newSimple_mem [DecidableEq ι] : |
| P ((DerivedCategory.singleFunctor (ModuleCat (T χ)) 0).obj (newSimple χ i)) := by |
| apply P.prop_of_iso (regularDerivedIso χ i) |
| apply DerivedSupport.prop_of_boundedAbove_terms P _ 0 |
| · intro j hj |
| exact CochainComplex.isZero_of_isStrictlyLE _ 0 j hj |
| · intro j |
| by_cases hj : ∃ n : ℕ, ComplexShape.embeddingDownNat.f n = j |
| · obtain ⟨n, hn⟩ := hj |
| apply P.prop_of_iso ((DerivedCategory.singleFunctor (ModuleCat (T χ)) 0).mapIso |
| ((periodicRegular χ i).extendXIso _ hn).symm) |
| exact newRegular_mem χ i P hI |
| · apply P.prop_of_isZero |
| apply (DerivedCategory.singleFunctor (ModuleCat (T χ)) 0).map_isZero |
| exact isZero_extend_X _ _ j (fun n hn => hj ⟨n,hn⟩) |
|
|
| omit hI [HasDerivedCategory.{w} (ModuleCat.{u} (T χ))] in |
| lemma injectivesGenerate [Fintype ι] (hχ : Function.Surjective χ) |
| (hnil : ∃ n : ℕ, (RingHom.ker χ.toRingHom)^n = ⊥) : |
| InjectivesGenerate (T χ) := by |
| classical |
| let := HasDerivedCategory.standard (ModuleCat.{u} (T χ)) |
| intro P hP hT hcop hI X |
| let := hP |
| let := hT |
| let (I : Type u) := hcop I |
| let : P.IsClosedUnderColimitsOfShape (Discrete ℕ) := |
| ObjectProperty.IsClosedUnderColimitsOfShape.of_equivalence |
| (Discrete.equivalence (Equiv.ulift : ULift.{u} ℕ ≃ ℕ)) |
| apply DerivedSupport.localizing_eq_top_of_all_modules P |
| · intro M |
| apply DerivedSupport.prop_all_of_product_augmentation P hcop |
| (augmentation χ).toRingHom (augmentation_surjective χ hχ) |
| (augmentation_ker_nilpotent χ hnil) _ M |
| intro j |
| cases j with |
| | inl i => exact oldSimple_mem χ i P hI |
| | inr i => exact newSimple_mem χ i P hI |
|
|
| end LittleFinitistic.CummingsConstruction |
|
|
| end |
|
|
| end OAI |
|
|
