| import OAI.Algebra.Finitistic.Bimodule.BoundedBimoduleDerived |
| import OAI.Algebra.Finitistic.Bimodule.BimoduleScalarEvaluation |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory CategoryTheory.Limits |
| namespace LittleFinitistic.BimoduleTotal |
| open ExternalTensor.Section4 |
| universe u |
| variable {k B : Type u} [Field k] [Ring B] [Algebra k B] |
|
|
| /-- Build the complete derived-tensor model from the actual native rectification, with no tensor-comparison assumption. -/ |
| def inducedBoundedProjective |
| {A J R : Type} [Fintype A] [Fintype J] [Fintype R] |
| (s : Quadratic k B (A := A) (J := J) (R := R)) |
| (D₀ D₁ D₂ : RightComplex B) (f : A → (D₀ ⟶ D₁)) (g : J → (D₁ ⟶ D₂)) |
| (H : ∀ r, Homotopy (relationMap s D₀ D₁ D₂ f g r) 0) |
| (lo hi : ℤ) |
| (b₀ : ∀ n, n < lo ∨ hi < n → IsZero (D₀.X n)) |
| (b₁ : ∀ n, n < lo ∨ hi < n → IsZero (D₁.X n)) |
| (b₂ : ∀ n, n < lo ∨ hi < n → IsZero (D₂.X n)) |
| (h₀ : ∀ n, Module.Projective Bᵐᵒᵖ (D₀.X n)) |
| (h₁ : ∀ n, Module.Projective Bᵐᵒᵖ (D₁.X n)) |
| (h₂ : ∀ n, Module.Projective Bᵐᵒᵖ (D₂.X n)) : |
| BoundedProjective k B (InducedRectification.total s D₀ D₁ D₂ f g H) where |
| lower := lo-2 |
| upper := hi |
| bounded := InducedRectification.total_bounded s D₀ D₁ D₂ f g H lo hi b₀ b₁ b₂ |
| projective := induced_term_projective s D₀ D₁ D₂ f g H h₀ h₁ h₂ |
|
|
| namespace BoundedProjective |
| variable {P : BiComplex k B} (data : BoundedProjective k B P) |
| variable [HasDerivedCategory.{u+1} (ModuleCat.{u} B)] |
|
|
| /-- The exact native module-valued degree-zero evaluation, using the module's original compatible k action. -/ |
| def nativeEvaluation (N : Type u) [AddCommGroup N] [Module k N] [Module B N] [IsScalarTower k B N] : |
| data.derived.obj (DerivedCategory.Q.obj (singleInput B (ModuleCat.of B N))) ≅ |
| DerivedCategory.Q.obj (((BimoduleBalancedTensor.functor k B N).mapHomologicalComplex (.up ℤ)).obj P) := |
| data.singleEvaluation (ModuleCat.of B N) ≪≫ DerivedCategory.Q.mapIso (scalarEvaluationIso k B N P) |
| end BoundedProjective |
| end LittleFinitistic.BimoduleTotal |
|
|
| end |
|
|
| end OAI |
|
|
