| import OAI.Algebra.Finitistic.Ordinary.OrdinaryFiniteConeReindex | |
| namespace OAI | |
| noncomputable section | |
| open CategoryTheory | |
| namespace LittleFinitistic.FiniteColumnComplex | |
| universe u | |
| variable {R : Type u} [Ring R] {l : ℕ} | |
| variable (K L : Fin l → CochainComplex (ModuleCat.{u} R) ℤ) | |
| variable (δ : Differential K) (ε : Differential L) | |
| variable (f : ∀ i, K i ⟶ L i) | |
| variable (hc : ∀ n, | |
| (ε.d n).comp (componentMap K L f n) = (componentMap K L f (n+1)).comp (δ.d n)) | |
| abbrev coneSource (n : ℤ) (z : term (coneColumn K L f) n) : term K (n+1) := | |
| fun i => (ConeCoordinates.coordinateEquiv (f i) n (z i)).1 | |
| abbrev coneTarget (n : ℤ) (z : term (coneColumn K L f) n) : term L n := | |
| fun i => (ConeCoordinates.coordinateEquiv (f i) n (z i)).2 | |
| theorem coneEquiv_symm_pair (n : ℤ) (z : term (coneColumn K L f) n) : | |
| ConeCoordinates.coordinateEquiv (complexMap K L δ ε f hc) n | |
| ((coneEquiv K L δ ε f hc n).symm z) = (coneSource K L f n z,coneTarget K L f n z) := by | |
| apply Prod.ext | |
| · funext i | |
| exact congrArg Prod.fst (coneEquiv_symm_coordinate K L δ ε f hc n z i) | |
| · funext i | |
| exact congrArg Prod.snd (coneEquiv_symm_coordinate K L δ ε f hc n z i) | |
| /-- The conjugated genuine cone differential has the expected finite-column formula. -/ | |
| theorem coneDifferential_coordinate (n : ℤ) (z : term (coneColumn K L f) n) (i : Fin l) : | |
| ConeCoordinates.coordinateEquiv (f i) (n+1) (coneDifferential K L δ ε f hc n z i) = | |
| (-(δ.d (n+1) (coneSource K L f n z)) i, | |
| ((f i).f (n+1)).hom (coneSource K L f n z i) + (ε.d n (coneTarget K L f n z)) i) := by | |
| change ConeCoordinates.coordinateEquiv (f i) (n+1) | |
| (coneEquiv K L δ ε f hc (n+1) | |
| (((totalCone K L δ ε f hc).d n (n+1)).hom ((coneEquiv K L δ ε f hc n).symm z)) i) = _ | |
| rw [coneEquiv_coordinate,ConeCoordinates.differential_fst,ConeCoordinates.differential_snd, | |
| coneEquiv_symm_pair] | |
| simp only [complex_d_hom] | |
| rfl | |
| /-- The cone of a componentwise total map is triangular with the actual column cones on its diagonal. -/ | |
| def coneTriangular : Differential (coneColumn K L f) where | |
| d := coneDifferential K L δ ε f hc | |
| square := coneDifferential_square K L δ ε f hc | |
| triangular n i z hz := by | |
| apply (ConeCoordinates.coordinateEquiv (f i) (n+1)).injective | |
| rw [coneDifferential_coordinate] | |
| have hSource : ∀ j : Fin l, j.val < i.val → coneSource K L f n z j = 0 := by | |
| intro j hj | |
| simp only [coneSource,hz j hj,map_zero,Prod.fst_zero] | |
| have hTarget : ∀ j : Fin l, j.val < i.val → coneTarget K L f n z j = 0 := by | |
| intro j hj | |
| simp only [coneTarget,hz j hj,map_zero,Prod.snd_zero] | |
| rw [δ.triangular (n+1) i _ hSource,ε.triangular n i _ hTarget] | |
| apply Prod.ext | |
| · exact (ConeCoordinates.differential_fst (f i) n (z i)).symm | |
| · exact (ConeCoordinates.differential_snd (f i) n (z i)).symm | |
| /-- The triangular cone model is isomorphic to the native cone, not an assumed cone replacement. -/ | |
| def coneIso : totalCone K L δ ε f hc ≅ complex (coneColumn K L f) (coneTriangular K L δ ε f hc) := | |
| HomologicalComplex.Hom.isoOfComponents (fun n => (coneEquiv K L δ ε f hc n).toModuleIso) (by | |
| intro n m h | |
| change n+1=m at h | |
| subst m | |
| rw [complex_d] | |
| apply ModuleCat.hom_ext | |
| apply LinearMap.ext | |
| intro z | |
| change coneEquiv K L δ ε f hc (n+1) | |
| (((totalCone K L δ ε f hc).d n (n+1)).hom | |
| ((coneEquiv K L δ ε f hc n).symm (coneEquiv K L δ ε f hc n z))) = | |
| coneEquiv K L δ ε f hc (n+1) (((totalCone K L δ ε f hc).d n (n+1)).hom z) | |
| rw [LinearEquiv.symm_apply_apply]) | |
| /-- Quasi-isomorphisms in every column induce a quasi-isomorphism of the actual finite totals. -/ | |
| theorem complexMap_quasiIso [∀ i, QuasiIso (f i)] : QuasiIso (complexMap K L δ ε f hc) := by | |
| apply (LittleFinitistic.quasiIso_iff_mappingCone_acyclic _).mpr | |
| have ha := acyclic (coneColumn K L f) (coneTriangular K L δ ε f hc) | |
| (fun i => (LittleFinitistic.quasiIso_iff_mappingCone_acyclic (f i)).mp inferInstance) | |
| intro n | |
| exact (exactAt_iff_of_quasiIsoAt (coneIso K L δ ε f hc).hom n).mpr (ha n) | |
| end LittleFinitistic.FiniteColumnComplex | |
| end | |
| end OAI | |