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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 | import OAI.Algebra.Finitistic.Rectification.RectificationTotal
import OAI.Algebra.Finitistic.Rectification.ColumnFinite
namespace OAI
open CategoryTheory CategoryTheory.Limits HomologicalComplex CochainComplex
noncomputable section
namespace ExternalTensor.Section4
universe u
variable {k : Type u} [Field k]
section General
variable {C : Type*} [Category C] [Preadditive C] [HasBinaryBiproducts C]
variable {L₀ L₁ L₂ : CochainComplex C ℤ} (p : L₁ ⟶ L₀) (q : L₂ ⟶ L₁)
(h : Homotopy (q ≫ p) 0)
/-- The rectification total has finite-dimensional terms whenever
an additive scalar restriction makes all three columns finite-dimensional. -/
theorem total_finite_after (F : C ⥤ ModuleCat.{u} k) [F.Additive]
(h₀ : ∀ i, FiniteDimensional k (F.obj (L₀.X i)))
(h₁ : ∀ i, FiniteDimensional k (F.obj (L₁.X i)))
(h₂ : ∀ i, FiniteDimensional k (F.obj (L₂.X i))) (i : ℤ) :
FiniteDimensional k (F.obj ((LittleFinitistic.RectificationTotal.total p q h).X i)) := by
let := preservesBinaryBiproducts_of_preservesBiproducts F
apply finite_of_iso k (F.mapIso (LittleFinitistic.RectificationTotal.termIso p q h i) ≪≫
F.mapBiprod _ _ ≪≫ biprod.mapIso (F.mapBiprod _ _) (Iso.refl _))
exact biprod_finite k (biprod_finite k (h₂ (i+2)) (h₁ (i+1))) (h₀ i)
end General
variable {B : Type u} [Ring B] [Algebra k B]
variable {A J R : Type} [Fintype A] [Fintype J] [Fintype R]
variable (s : Quadratic k B (A := A) (J := J) (R := R))
variable (D₀ D₁ D₂ : RightComplex B)
variable (f : A → (D₀ ⟶ D₁)) (g : J → (D₁ ⟶ D₂))
variable (H : ∀ r, Homotopy (relationMap s D₀ D₁ D₂ f g r) 0)
/-- The finite-dimensionality conclusion for the manuscript's constructed P,
with no projectivity or finite-dimensionality assumption on P or its columns. -/
theorem induced_total_finite [FiniteDimensional k B]
(h₀ : ∀ i, Module.Finite Bᵐᵒᵖ (D₀.X i))
(h₁ : ∀ i, Module.Finite Bᵐᵒᵖ (D₁.X i))
(h₂ : ∀ i, Module.Finite Bᵐᵒᵖ (D₂.X i)) (i : ℤ) :
FiniteDimensional k ((forgetK k B).obj
((LittleFinitistic.RectificationTotal.total (p s D₀ D₁ D₂ f g)
(q s D₀ D₁ D₂ f g) (nullHomotopy s D₀ D₁ D₂ f g H)).X i)) := by
exact total_finite_after _ _ _ (forgetK k B)
(fun n => (columns_finite s D₀ D₁ D₂ f g h₀ h₁ h₂ n).1)
(fun n => (columns_finite s D₀ D₁ D₂ f g h₀ h₁ h₂ n).2.1)
(fun n => (columns_finite s D₀ D₁ D₂ f g h₀ h₁ h₂ n).2.2) i
end ExternalTensor.Section4
end
end OAI
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