| import OAI.Algebra.Finitistic.Bar.GlobalIndexedComparison |
| import OAI.Algebra.Finitistic.Bar.IntegerDerivedMultibar |
| import OAI.Algebra.Finitistic.Bar.StrictSupportExtension |
|
|
| namespace OAI |
| open CategoryTheory CategoryTheory.Limits TensorProduct TrivSqZeroExt HomologicalComplex |
| open scoped TensorProduct DirectSum |
| noncomputable section |
| namespace DerivedBar.ParityBridge |
| open DerivedBar.Multibar DerivedBar.RecursiveTotal |
| 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 integerCountRestrictionScalarModule (V : ModuleCat D) : Module k V := |
| BimoduleTensor.scalarModule k D V |
| local instance integerCountRestrictionScalarTower (V : ModuleCat D) : IsScalarTower k D V := |
| IsScalarTower.of_algebraMap_smul (fun _ _ => rfl) |
| local instance integerCountRestrictionIndexDecidableEq (r n : ℕ) : |
| DecidableEq (Index r n) := Classical.decEq _ |
| local instance integerCountRestrictionShapeDecidableEq (n r : ℕ) : |
| DecidableEq (Multibar.Indexed.Shape n r) := Classical.decEq _ |
| variable (N : ModuleCat 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 instance integerCountRestrictionInflatedModule : Module A N := inflatedModule D X N |
| local instance integerCountRestrictionInflatedScalarTower : IsScalarTower k A N := |
| IsScalarTower.of_algebraMap_smul |
| (fun a v => IsScalarTower.algebraMap_smul D a v) |
|
|
| local notation "C" => ModuleCat D |
| local notation "E" => ComplexShape.embeddingDownNat |
| local notation "K" => Multibar.Indexed.signedBlockComplex k D X N (inflated_ideal_zero D X N) |
|
|
| /-- The actual integer cochain count complex shifted by -r. -/ |
| def deshiftedCount (r : ℕ) : CochainComplex C ℤ := |
| ((integerExtension D).obj (K r))⟦-(r:ℤ)⟧ |
|
|
| def deshiftedCountRestriction (r : ℕ) : ChainComplex C ℕ := |
| (deshiftedCount k D X N r).restriction E |
|
|
| def deshiftedCountXIso (r n : ℕ) : (deshiftedCountRestriction k D X N r).X n ≅ |
| indexedModule k D X N r n := |
| (K r).extendXIso E |
| (show (E).f (r+n) = (E).f n + -(r:ℤ) by change -((r+n:ℕ):ℤ) = -(n:ℤ) + -(r:ℤ); push_cast; omega) |
|
|
| theorem units_hom_apply (V W : ModuleCat.{u} D) (s : ℤˣ) (f : V ⟶ W) (v : V) : |
| (s • f).hom v = s • f.hom v := by |
| change ((s:ℤ) • f).hom v = (s:ℤ) • f.hom v |
| simp only [ModuleCat.hom_smul, LinearMap.smul_apply] |
|
|
| /-- The standard unit-valued cochain shift sign is the proved count sign. -/ |
| theorem deshiftedCount_sign (r n : ℕ) : |
| (-(r:ℤ)).negOnePow • (K r).d (r+(n+1)) (r+n) = indexedDifferential k D X N r n := by |
| have hd : (K r).d (r+(n+1)) (r+n) = |
| ModuleCat.ofHom (Multibar.Indexed.blockDifferential k D X N (inflated_ideal_zero D X N) (r+n) r) := by |
| change ChainComplex.of.d |
| (fun m => ModuleCat.of D (⨁ p : Multibar.Indexed.Shape m r, Multibar.Indexed.outerTensor k D X N p)) |
| (fun m => ModuleCat.ofHom (Multibar.Indexed.blockDifferential k D X N |
| (inflated_ideal_zero D X N) m r)) ((r+n)+1) (r+n) = _ |
| exact ChainComplex.of_d _ _ (r+n) |
| rw [hd] |
| apply ModuleCat.hom_ext |
| apply LinearMap.ext |
| intro v |
| erw [units_hom_apply] |
| erw [globalIndexedComparison] |
| change (-(r:ℤ)).negOnePow • |
| Multibar.Indexed.blockDifferential k D X N (inflated_ideal_zero D X N) (r+n) r v = |
| (-1:k)^r • |
| Multibar.Indexed.blockDifferential k D X N (inflated_ideal_zero D X N) (r+n) r v |
| have hu : (r:ℤ).negOnePow = (-1:ℤˣ)^r := by |
| apply Units.ext |
| simpa using Int.coe_negOnePow_natCast r |
| rw [Int.negOnePow_neg, hu] |
| exact totalSign_smul k (⨁ p : Multibar.Indexed.Shape (r+n) r, Multibar.Indexed.outerTensor k D X N p) r |
| (Multibar.Indexed.blockDifferential k D X N (inflated_ideal_zero D X N) (r+n) r v) |
|
|
| theorem deshiftedCount_d (r n : ℕ) : |
| (deshiftedCountRestriction k D X N r).d (n+1) n = |
| (deshiftedCountXIso k D X N r (n+1)).hom ≫ indexedDifferential k D X N r n ≫ |
| (deshiftedCountXIso k D X N r n).inv := by |
| change (-(r:ℤ)).negOnePow • ((K r).extend E).d ((E).f (n+1) + -(r:ℤ)) ((E).f n + -(r:ℤ)) = _ |
| rw [extend_d_eq (K r) E |
| (show (E).f (r+(n+1))=(E).f (n+1) + -(r:ℤ) by change -((r+(n+1):ℕ):ℤ) = -((n+1:ℕ):ℤ) + -(r:ℤ); push_cast; omega) |
| (show (E).f (r+n)=(E).f n + -(r:ℤ) by change -((r+n:ℕ):ℤ) = -(n:ℤ) + -(r:ℤ); push_cast; omega)] |
| rw [← deshiftedCount_sign] |
| erw [CategoryTheory.Linear.units_smul_comp, CategoryTheory.Linear.comp_units_smul] |
| rfl |
|
|
| /-- A genuine chain isomorphism using the standard integer shift differential, |
| not a newly stipulated shift or transported count differential. -/ |
| def indexedDeshiftedCountIso (r : ℕ) : indexedMultibar k D X N r ≅ |
| deshiftedCountRestriction k D X N r := |
| HomologicalComplex.Hom.isoOfComponents |
| (fun n => (deshiftedCountXIso k D X N r n).symm) (by |
| intro i j hij |
| change j+1=i at hij |
| subst i |
| rw [deshiftedCount_d] |
| simp only [Iso.symm_hom] |
| erw [Iso.inv_hom_id_assoc] |
| simp only [indexedMultibar, ChainComplex.of_d]) |
| end DerivedBar.ParityBridge |
|
|
| end |
|
|
| end OAI |
|
|
