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ae00908 | 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 89 90 91 92 93 94 95 96 97 98 99 100 101 | import OAI.Algebra.Finitistic.Rectification.SourceFiltrationExact
namespace OAI
open CategoryTheory CategoryTheory.Limits HomologicalComplex CochainComplex
open LittleFinitistic.RectificationTotal
noncomputable section
namespace ExternalTensor.Section4.SourceFiltration
universe u
variable {k B : Type u} [Field k] [Ring B] [Algebra k B]
variable {A J R : Type} [Fintype A] [Fintype J] [Fintype R]
variable (t : AssemblyData (C := BiComplex k B) (A := A) (J := J) (R := R))
@[reassoc (attr := simp)]
theorem inr_prod {S : Type u} [Ring S] (X Y : ModuleCat.{u} S) :
(biprod.inr : Y ⟶ X ⊞ Y) ≫ (ModuleCat.biprodIsoProd X Y).hom =
ModuleCat.ofHom (LinearMap.inr S X Y) := by
apply (cancel_mono (ModuleCat.biprodIsoProd X Y).inv).1
simp only [Category.assoc, Iso.hom_inv_id, Category.comp_id]
apply biprod.hom_ext
· simp only [Category.assoc, ModuleCat.biprodIsoProd_inv_comp_fst, biprod.inr_fst]
rfl
· simp only [Category.assoc, ModuleCat.biprodIsoProd_inv_comp_snd, biprod.inr_snd]
rfl
omit [Fintype R] in
/-- The canonical zero-column inclusion has exactly its original term coordinate. -/
theorem inclusion_product (n : ℤ) :
(inclusion (p2 t) 0 (h2 t)).f n ≫ (totalTermIso (p2 t) 0 (h2 t) n).hom =
ModuleCat.ofHom (LinearMap.inr (Enveloping k B)
(Z.X (n+2) × Z.X (n+1)) (t.V₂.X n)) := by
dsimp only [totalTermIso, termIso, LittleFinitistic.RectificationTotal.inclusion]
erw [Iso.trans_hom, Iso.trans_hom, Iso.trans_hom,
biprod.mapIso_hom, Iso.refl_hom, biprod.mapIso_hom, Iso.refl_hom]
erw [← Category.assoc, ← Category.assoc, ← Category.assoc,
cone_inr_termIso, biprod.inr_map, Category.id_comp, biprod.inr_map, Category.id_comp]
exact inr_prod _ _
omit [Fintype R] in
/-- Removing the two zero columns preserves the actual V2 inclusion. -/
theorem inclusion2_bijective (n : ℤ) :
Function.Bijective ((inclusion (p2 t) 0 (h2 t)).f n) := by
let e := (totalTermIso (p2 t) 0 (h2 t) n).toLinearEquiv
have he (x : t.V₂.X n) : e ((inclusion (p2 t) 0 (h2 t)).f n x) = ((0,0),x) :=
congrArg (fun a => a x) (inclusion_product t n)
refine ⟨?_, ?_⟩
· intro x y hxy
have := congrArg e hxy
simpa only [he] using congrArg Prod.snd this
· intro y
refine ⟨(e y).2, e.injective ?_⟩
rw [he]
exact Prod.ext (Subsingleton.elim _ _) rfl
instance inclusion2_isIso : IsIso (inclusion (p2 t) 0 (h2 t)) := by
have (n : ℤ) : IsIso ((inclusion (p2 t) 0 (h2 t)).f n) :=
(ConcreteCategory.isIso_iff_bijective _).2 (inclusion2_bijective t n)
exact HomologicalComplex.Hom.isIso_of_components _
def v2Iso : t.V₂ ≅ total (p2 t) 0 (h2 t) := asIso (inclusion (p2 t) 0 (h2 t))
/-- F>=2 really is V2, embedded by its original vertex summand inclusion. -/
def v2Inclusion : t.V₂ ⟶ total (p1 t) 0 (h1 t) :=
biprod.inr ≫ inclusion (p1 t) 0 (h1 t)
omit [Fintype R] in
theorem v2Inclusion_eq : v2Inclusion t = (v2Iso t).hom ≫ (include2 t).totalMap := by
symm
exact (include2 t).inclusion_totalMap
omit [Fintype R] in
/-- The source-one sequence with V2 itself as its first term. -/
theorem v2_shortExact (n : ℤ) :
Function.Injective ((v2Inclusion t).f n) ∧
Function.Exact ((v2Inclusion t).f n) ((quotient1 t).totalMap.f n) ∧
Function.Surjective ((quotient1 t).totalMap.f n) := by
have E := source1_shortExact t n
have I := inclusion2_bijective t n
rw [v2Inclusion_eq]
refine ⟨E.1.comp I.1, ?_, E.2.2⟩
intro y
constructor
· intro hy
obtain ⟨x, hx⟩ := (E.2.1 y).mp hy
obtain ⟨z, rfl⟩ := I.2 x
exact ⟨z, hx⟩
· rintro ⟨x, rfl⟩
exact E.2.1.apply_apply_eq_zero _
variable (h : Homotopy (t.q ≫ t.p) 0)
/-- The composite filtration inclusion is the original V2-to-total map. -/
theorem v2_nested : v2Inclusion t ≫ (include1 t h).totalMap =
t.in₂ ≫ inclusion t.p t.q h := by
dsimp only [v2Inclusion]
rw [Category.assoc, (include1 t h).inclusion_totalMap]
rfl
end ExternalTensor.Section4.SourceFiltration
end
end OAI
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