| import OAI.Algebra.Finitistic.Bar.ModuleMultibarCoordinates |
|
|
| namespace OAI |
| open CategoryTheory CategoryTheory.Limits CategoryTheory.Preadditive TensorProduct |
| open scoped TensorProduct DirectSum |
| noncomputable section |
| namespace DerivedBar.RecursiveTotal |
| open DerivedBar.Multibar |
| 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 (N : ModuleCat.{u} D) : Module k N := BimoduleTensor.scalarModule k D N |
| local instance (N : ModuleCat.{u} D) : IsScalarTower k D N := |
| IsScalarTower.of_algebraMap_smul (fun _ _ => rfl) |
| variable (N : ModuleCat.{u} D) |
| local notation "Z" => (Functor.obj (ChainComplex.single₀ (ModuleCat D)) N) |
| local notation "B" => leadingBarSequence k D X |
| local notation "F" => flatLeadingMultibar k D X Z |
|
|
| abbrev indexedModule (r n : ℕ) : ModuleCat.{u} D := |
| ModuleCat.of D (⨁ p : Multibar.Indexed.Shape (r+n) r, |
| Multibar.Indexed.outerTensor k D X N p) |
|
|
| /-- The independently recursive multibar faces in the exact Shape coordinates. |
| This uses the horizontal/vertical differential of RecursiveTotal, not the count |
| complex differential. -/ |
| def indexedDifferential (r n : ℕ) : indexedModule k D X N r (n+1) ⟶ indexedModule k D X N r n := |
| (moduleFlatCoordinates k D X N r (n+1)).inv ≫ |
| (F r).d (n+1) n ≫ (moduleFlatCoordinates k D X N r n).hom |
|
|
| theorem indexedDifferential_sq (r n : ℕ) : |
| indexedDifferential k D X N r (n+1) ≫ indexedDifferential k D X N r n = 0 := by |
| simp only [indexedDifferential, Category.assoc, Iso.hom_inv_id_assoc] |
| rw [← Category.assoc ((F r).d (n+2) (n+1)), HomologicalComplex.d_comp_d, |
| zero_comp, comp_zero] |
|
|
| /-- Exact tensor-block coordinates, with the unshifted multibar total degree n. -/ |
| def indexedMultibar (r : ℕ) : ChainComplex (ModuleCat.{u} D) ℕ := |
| ChainComplex.of (indexedModule k D X N r) (indexedDifferential k D X N r) |
| (indexedDifferential_sq k D X N r) |
|
|
| /-- Chain isomorphism from the exact tensor-block direct sum to the recursive flat model. -/ |
| def indexedFlatIso (r : ℕ) : indexedMultibar k D X N r ≅ F r := |
| HomologicalComplex.Hom.isoOfComponents |
| (fun n => (moduleFlatCoordinates k D X N r n).symm) |
| (by |
| intro i j hij |
| change j+1=i at hij |
| subst i |
| change (moduleFlatCoordinates k D X N r (j+1)).inv ≫ (F r).d (j+1) j = |
| (indexedMultibar k D X N r).d (j+1) j ≫ (moduleFlatCoordinates k D X N r j).inv |
| simp only [indexedMultibar, ChainComplex.of_d, indexedDifferential, |
| Category.assoc, Iso.hom_inv_id, Category.comp_id]) |
|
|
| /-- Actual chain isomorphism flattening the recursive leading total into the |
| same Shape-indexed block tensors as the square-zero count decomposition. -/ |
| def indexedLeadingIso (r : ℕ) : indexedMultibar k D X N r ≅ |
| barTotal k D (explicitMultibar k D X r Z) := |
| indexedFlatIso k D X N r ≪≫ flatLeadingMultibarIso k D X Z r |
|
|
| /-- Exact signed differential formula on a recursive generator, now in the |
| literal Shape-indexed block coordinates. The vertical face includes (-1)^i. -/ |
| theorem indexedDifferential_on_generator (r n i q : ℕ) (h : i+q=n+1) (t : Index r q) : |
| lof (B r) Z (r+1) (n+1) ⟨⟨⟨i,q⟩,h⟩,t⟩ ≫ |
| (moduleFlatCoordinates k D X N r (n+1)).hom ≫ indexedDifferential k D X N r n = |
| (horizontal (B r) Z r n i q h t + |
| vertical (B r) Z r n i q h t (differential (B r) Z r)) ≫ |
| (moduleFlatCoordinates k D X N r n).hom := by |
| dsimp only [indexedDifferential] |
| erw [Iso.hom_inv_id_assoc] |
| simp only [flatLeadingMultibar, flatComplex, ChainComplex.of_d] |
| change lof (B r) Z (r+1) (n+1) _ ≫ differential (B r) Z (r+1) n ≫ _ = _ |
| rw [lof_differential_succ_assoc] |
| rfl |
|
|
| /-- Genuine derived tensor iteration represented by the exact indexed block |
| complex. No count splitting or projective-dimension statement is assumed. -/ |
| def derivedIterateIndexedIso (r : ℕ) : |
| (derivedTensorIterate k D X r).obj ((chainQ D).obj Z) ≅ |
| (chainQ D).obj (indexedMultibar k D X N r) := |
| derivedIterateFlatIso k D X Z r ≪≫ (chainQ D).mapIso (indexedFlatIso k D X N r).symm |
| end DerivedBar.RecursiveTotal |
|
|
| end |
|
|
| end OAI |
|
|
