| import Mathlib |
| import OAI.Algebra.FinitisticAsymmetry.Triangular.InjectiveCorners |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open CategoryTheory |
| namespace LittleFinitistic.CummingsConstruction |
| open CategoryTheory CategoryTheory.Limits HomologicalComplex |
| universe u |
| variable {k C ι : Type u} [Field k] [Ring C] [Algebra k C] |
| variable (χ : C →ₐ[k] (ι → k)) (i : ι) |
|
|
| def oldCharacter : T χ →+* k := |
| (Pi.evalRingHom (fun _ : Sum ι ι => k) (.inl i)).comp (augmentation χ).toRingHom |
|
|
| def newCharacter : T χ →+* k := |
| (Pi.evalRingHom (fun _ : Sum ι ι => k) (.inr i)).comp (augmentation χ).toRingHom |
|
|
| def newProjection : T χ →+* DualNumber k where |
| toFun r := r.fst.2 i |
| map_one' := rfl |
| map_zero' := rfl |
| map_add' _ _ := rfl |
| map_mul' _ _ := rfl |
|
|
| abbrev oldSimple := (ModuleCat.restrictScalars (oldCharacter χ i)).obj (ModuleCat.of k k) |
| abbrev newSimple := (ModuleCat.restrictScalars (newCharacter χ i)).obj (ModuleCat.of k k) |
| abbrev newRegular := (ModuleCat.restrictScalars (newProjection χ i)).obj (ModuleCat.of (DualNumber k) (DualNumber k)) |
| abbrev newInjectiveObj := ModuleCat.of (T χ) (NewInjective χ i) |
|
|
| def newRegularι : newRegular χ i ⟶ newInjectiveObj χ i := ModuleCat.ofHom |
| (X := newRegular χ i) (Y := newInjectiveObj χ i) (show |
| newRegular χ i →ₗ[T χ] newInjectiveObj χ i from { |
| toFun := fun x => (0,x.fst,x.snd) |
| map_add' := fun x y => by |
| apply NewInjective.ext (χ := χ) (i := i) |
| · exact (zero_add (0 : k)).symm |
| · rfl |
| · rfl |
| map_smul' := fun r x => by |
| apply NewInjective.ext (χ := χ) (i := i) |
| · change (0 : k) = cval χ i r * 0 |
| simp |
| · rfl |
| · change ehead χ i r*x.snd + etail χ i r*x.fst = |
| oval χ i r*0 + etail χ i r*x.fst + ehead χ i r*x.snd |
| ring }) |
|
|
| def newInjectiveπ : newInjectiveObj χ i ⟶ oldSimple χ i := ModuleCat.ofHom |
| (X := newInjectiveObj χ i) (Y := oldSimple χ i) (show |
| newInjectiveObj χ i →ₗ[T χ] oldSimple χ i from { |
| toFun := fun x => x.1 |
| map_add' := fun _ _ => rfl |
| map_smul' := fun _ _ => rfl }) |
|
|
| def newRegularπ : newRegular χ i ⟶ newSimple χ i := ModuleCat.ofHom |
| (X := newRegular χ i) (Y := newSimple χ i) (show |
| newRegular χ i →ₗ[T χ] newSimple χ i from { |
| toFun := fun x => x.fst |
| map_add' := fun _ _ => rfl |
| map_smul' := fun _ _ => rfl }) |
|
|
| def newSimpleι : newSimple χ i ⟶ newRegular χ i := ModuleCat.ofHom |
| (X := newSimple χ i) (Y := newRegular χ i) (show |
| newSimple χ i →ₗ[T χ] newRegular χ i from { |
| toFun := fun a => (0,a) |
| map_add' := fun a b => by |
| apply TrivSqZeroExt.ext |
| · exact (zero_add (0 : k)).symm |
| · rfl |
| map_smul' := fun r a => by |
| apply TrivSqZeroExt.ext |
| · change (0 : k) = ehead χ i r * 0 |
| simp |
| · change ehead χ i r*(show k from a) = ehead χ i r*(show k from a) + etail χ i r*0 |
| simp }) |
|
|
| def newDifferential : newInjectiveObj χ i ⟶ newInjectiveObj χ i := ModuleCat.ofHom |
| (X := newInjectiveObj χ i) (Y := newInjectiveObj χ i) (show |
| newInjectiveObj χ i →ₗ[T χ] newInjectiveObj χ i from { |
| toFun := fun x => (0,0,x.2.1) |
| map_add' := fun x y => by |
| apply NewInjective.ext (χ := χ) (i := i) |
| · exact (zero_add (0 : k)).symm |
| · exact (zero_add (0 : k)).symm |
| · rfl |
| map_smul' := fun r x => by |
| apply NewInjective.ext (χ := χ) (i := i) |
| · change (0 : k) = cval χ i r * 0 |
| simp |
| · change (0 : k) = ehead χ i r * 0 |
| simp |
| · change ehead χ i r*x.2.1 = oval χ i r*0 + etail χ i r*0 + ehead χ i r*x.2.1 |
| simp }) |
|
|
| def epsDifferential : newRegular χ i ⟶ newRegular χ i := newRegularπ χ i ≫ newSimpleι χ i |
|
|
| @[simp] lemma newDifferential_sq : newDifferential χ i ≫ newDifferential χ i = 0 := by |
| apply ModuleCat.hom_ext |
| apply LinearMap.ext |
| intro x |
| rfl |
|
|
| @[simp] lemma epsDifferential_sq : epsDifferential χ i ≫ epsDifferential χ i = 0 := by |
| apply ModuleCat.hom_ext |
| apply LinearMap.ext |
| intro x |
| rfl |
|
|
| lemma newRegularι_comm : newRegularι χ i ≫ newDifferential χ i = |
| epsDifferential χ i ≫ newRegularι χ i := by |
| apply ModuleCat.hom_ext |
| apply LinearMap.ext |
| intro x |
| rfl |
|
|
| lemma newInjectiveπ_comm : newDifferential χ i ≫ newInjectiveπ χ i = 0 := by |
| apply ModuleCat.hom_ext |
| apply LinearMap.ext |
| intro x |
| rfl |
|
|
| lemma newRegularι_π : newRegularι χ i ≫ newInjectiveπ χ i = 0 := by |
| apply ModuleCat.hom_ext |
| apply LinearMap.ext |
| intro x |
| rfl |
|
|
| lemma newRegular_exact : |
| (ShortComplex.mk (newRegularι χ i) (newInjectiveπ χ i) (newRegularι_π χ i)).ShortExact where |
| mono_f := by |
| rw [ModuleCat.mono_iff_injective] |
| intro x y h |
| apply TrivSqZeroExt.ext |
| · exact congrArg (fun z : NewInjective χ i => z.2.1) h |
| · exact congrArg (fun z : NewInjective χ i => z.2.2) h |
| epi_g := by |
| rw [ModuleCat.epi_iff_surjective] |
| intro a |
| exact ⟨(a,0,0), rfl⟩ |
| exact := by |
| rw [ShortComplex.moduleCat_exact_iff] |
| intro x hx |
| refine ⟨(x.2.1,x.2.2), ?_⟩ |
| apply NewInjective.ext (χ := χ) (i := i) |
| · exact hx.symm |
| · rfl |
| · rfl |
|
|
| def periodicInjective : ChainComplex (ModuleCat (T χ)) ℕ := |
| ChainComplex.of (fun _ => newInjectiveObj χ i) (fun _ => newDifferential χ i) |
| (fun _ => newDifferential_sq χ i) |
|
|
| def periodicRegular : ChainComplex (ModuleCat (T χ)) ℕ := |
| ChainComplex.of (fun _ => newRegular χ i) (fun _ => epsDifferential χ i) |
| (fun _ => epsDifferential_sq χ i) |
|
|
| def periodicOld : ChainComplex (ModuleCat (T χ)) ℕ := |
| ChainComplex.of (fun _ => oldSimple χ i) (fun _ => 0) (fun _ => by simp) |
|
|
| def periodicι : periodicRegular χ i ⟶ periodicInjective χ i := |
| ChainComplex.ofHom (fun _ => newRegularι χ i) (fun n => by |
| simpa [periodicInjective, periodicRegular] using newRegularι_comm χ i) |
|
|
| def periodicπ : periodicInjective χ i ⟶ periodicOld χ i := |
| ChainComplex.ofHom (fun _ => newInjectiveπ χ i) (fun n => by |
| simpa [periodicInjective, periodicOld] using (newInjectiveπ_comm χ i).symm) |
|
|
| lemma periodicι_π : periodicι χ i ≫ periodicπ χ i = 0 := by |
| apply HomologicalComplex.hom_ext |
| intro n |
| exact newRegularι_π χ i |
|
|
| lemma periodic_shortExact : |
| (ShortComplex.mk (periodicι χ i) (periodicπ χ i) (periodicι_π χ i)).ShortExact where |
| mono_f := HomologicalComplex.mono_of_mono_f _ (fun _ => (newRegular_exact χ i).mono_f) |
| epi_g := HomologicalComplex.epi_of_epi_f _ (fun _ => (newRegular_exact χ i).epi_g) |
| exact := HomologicalComplex.exact_of_degreewise_exact _ (fun _ => (newRegular_exact χ i).exact) |
|
|
| lemma epsDifferential_π : epsDifferential χ i ≫ newRegularπ χ i = 0 := by |
| apply ModuleCat.hom_ext |
| apply LinearMap.ext |
| intro x |
| rfl |
|
|
| lemma epsDifferential_exact : |
| (ShortComplex.mk (epsDifferential χ i) (epsDifferential χ i) |
| (epsDifferential_sq χ i)).Exact := by |
| rw [ShortComplex.moduleCat_exact_iff] |
| intro x hx |
| have hx' : x.fst = 0 := congrArg TrivSqZeroExt.snd hx |
| refine ⟨(x.snd,0), ?_⟩ |
| apply TrivSqZeroExt.ext |
| · exact hx'.symm |
| · rfl |
|
|
| lemma epsDifferential_π_exact : |
| (ShortComplex.mk (epsDifferential χ i) (newRegularπ χ i) |
| (epsDifferential_π χ i)).Exact := by |
| rw [ShortComplex.moduleCat_exact_iff] |
| intro x hx |
| refine ⟨(x.snd,0), ?_⟩ |
| apply TrivSqZeroExt.ext |
| · exact hx.symm |
| · rfl |
|
|
| instance newRegularπ_epi : Epi (newRegularπ χ i) := by |
| rw [ModuleCat.epi_iff_surjective] |
| intro a |
| exact ⟨(a,0), rfl⟩ |
|
|
| def periodicAug : periodicRegular χ i ⟶ (ChainComplex.single₀ (ModuleCat (T χ))).obj |
| (newSimple χ i) := |
| (ChainComplex.toSingle₀Equiv _ _).symm ⟨newRegularπ χ i, epsDifferential_π χ i⟩ |
|
|
| instance periodicAug_quasiIso : QuasiIso (periodicAug χ i) := ⟨fun n => by |
| cases n with |
| | zero => |
| rw [ChainComplex.quasiIsoAt₀_iff, ShortComplex.quasiIso_iff_of_zeros'] |
| · exact ⟨epsDifferential_π_exact χ i, newRegularπ_epi χ i⟩ |
| all_goals rfl |
| | succ n => |
| rw [quasiIsoAt_iff_exactAt'] |
| · rw [HomologicalComplex.exactAt_iff' _ (n+2) (n+1) n (by simp) (by simp)] |
| have hd : ChainComplex.of.d (fun _ : ℕ => newRegular χ i) |
| (fun _ : ℕ => epsDifferential χ i) (n+2) (n+1) = epsDifferential χ i := |
| ChainComplex.of_d _ _ (n+1) |
| simpa only [HomologicalComplex.sc', HomologicalComplex.shortComplexFunctor', |
| periodicRegular, hd, ChainComplex.of_d] using epsDifferential_exact χ i |
| · apply ChainComplex.exactAt_succ_single_obj⟩ |
| |
| end LittleFinitistic.CummingsConstruction |
| |
| end |
| |
| end OAI |
| |
