| import OAI.Algebra.Finitistic.Bar.RecursiveSuffixCoefficient |
| import OAI.Algebra.Finitistic.Bar.LeadingBarSuffix |
|
|
| namespace OAI |
| open CategoryTheory CategoryTheory.Limits TensorProduct TrivSqZeroExt |
| open scoped TensorProduct DirectSum |
| noncomputable section |
| namespace DerivedBar.ParityBridge |
| universe u |
| variable (k D X : Type u) [Field k] [Ring D] [Algebra k D] |
| [AddCommGroup X] [Module k X] [Module D X] [Module Dᵐᵒᵖ X] |
| [SMulCommClass D Dᵐᵒᵖ X] [IsScalarTower k D X] [IsScalarTower k Dᵐᵒᵖ X] |
| local instance (V : ModuleCat.{u} D) : Module k V := BimoduleTensor.scalarModule k D V |
| local instance (V : ModuleCat.{u} D) : IsScalarTower k D V := |
| IsScalarTower.of_algebraMap_smul (fun _ _ => rfl) |
| variable (N : ModuleCat.{u} D) |
| local notation "A" => TrivSqZeroExt D X |
| local notation "XB" => BimoduleTensor.twoSidedBarFunctor k D X |
| local notation "Z" => Functor.obj (ChainComplex.single₀ (ModuleCat D)) N |
| local notation "B" => (fun _ : ℕ => XB) |
| local notation "c" => ComplexShape.down ℕ |
| local notation "I" => RecursiveTotal.iter B Z |
| local instance : Module A N := inflatedModule D X N |
| local instance : IsScalarTower k A N := IsScalarTower.of_algebraMap_smul |
| (fun a v => IsScalarTower.algebraMap_smul D a v) |
|
|
| /-- The exterior D bar is assembled on its actual finite totalization diagonal. -/ |
| def leadingTotalEmbedding (r n : ℕ) : |
| (barTotal k D (I r)).X n ⟶ ModuleCat.of D (D ⊗[k] term k A N (r+n)) := |
| totalMapDesc (𝟭 _) (barBicomplex k D (I r)) (fun i q h => |
| ModuleCat.ofHom ((outerCast k D X N (show (r+q)+i=r+n by omega)).comp |
| (outerBarSuffix k D X N ((I r).X q) (r+q) i (innerTotalEmbedding k D X N r q)))) |
|
|
| @[reassoc (attr := simp)] theorem leadingTotalEmbedding_ι (r n i q : ℕ) (h : i+q=n) : |
| (barBicomplex k D (I r)).ιTotal c i q n h ≫ leadingTotalEmbedding k D X N r n = |
| ModuleCat.ofHom ((outerCast k D X N (show (r+q)+i=r+n by omega)).comp |
| (outerBarSuffix k D X N ((I r).X q) (r+q) i (innerTotalEmbedding k D X N r q))) := by |
| exact ι_totalMapDesc (𝟭 _) (barBicomplex k D (I r)) _ i q h |
|
|
| /-- The independently recursive inner total is the previously constructed explicit multibar. -/ |
| theorem innerIter_eq_explicit (r : ℕ) : I r = explicitMultibar k D X r Z := by |
| induction r with |
| | zero => rfl |
| | succ r ih => |
| simp only [RecursiveTotal.iter, RecursiveTotal.bicomplex, ih, explicitMultibar, |
| BimoduleTensor.explicitBarTotal, BimoduleTensor.twoSidedBarBicomplex] |
| end DerivedBar.ParityBridge |
|
|
| end |
|
|
| end OAI |
|
|
