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216fb31 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 | 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
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