| import OAI.Algebra.Finitistic.Bar.LeadingTotalDifferential |
|
|
| namespace OAI |
| open CategoryTheory CategoryTheory.Limits CategoryTheory.Preadditive 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 "DB" => barFunctor k D |
| 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 notation "K" => barBicomplex k D |
| 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) |
|
|
| theorem outerCast_smul {a b : ℕ} (h : a=b) (z : k) |
| (v : D ⊗[k] term k A N a) : |
| outerCast k D X N h (z • v) = z • outerCast k D X N h v := by |
| subst b |
| simp only [outerCast_refl, LinearMap.id_apply] |
|
|
| theorem leadingTotalComparison_positive (r n i q : ℕ) (h : i+(q+1)=n+1) |
| (v : (((DB).obj ((I r).X (q+1))).X i)) : |
| SquareZero.multibarTwoSidedDifferential k D X N (r+n) |
| (outerCast k D X N (show (r+(q+1))+i=r+(n+1) by omega) |
| (outerBarSuffix k D X N ((I r).X (q+1)) (r+(q+1)) i |
| (innerTotalEmbedding k D X N r (q+1)) v)) = |
| (((K (I r)).d₁ c i (q+1) n ≫ leadingTotalEmbedding k D X N r n).hom v) + |
| (((K (I r)).d₂ c i (q+1) n ≫ leadingTotalEmbedding k D X N r n).hom v) := by |
| have hδ : (SquareZero.multibarAugmentedDifferential k D X N (r+q)).comp |
| (innerTotalEmbedding k D X N r (q+1)) = |
| -((innerTotalEmbedding k D X N r q).comp (((I r).d (q+1) q).hom.restrictScalars k)) := by |
| ext w |
| exact innerTotalEmbedding_differential k D X N r q w |
| cases i with |
| | zero => |
| have hn : n=q := by omega |
| subst n |
| simp only [Nat.add_zero, outerCast_refl, LinearMap.id_apply] |
| erw [parity_outerBarSuffix_zero k D X N ((I r).X (q+1)) (r+q) |
| (innerTotalEmbedding k D X N r (q+1)) |
| (fun w => innerTotalEmbedding_coefficient_of_pos k D X N r (q+1) (r+q) rfl w)] |
| rw [hδ, outerBarSuffix_neg, LinearMap.neg_apply, neg_neg] |
| rw [leadingTotalEmbedding_d₁_zero] |
| erw [leadingTotalEmbedding_d₂ k D X N r q 0 q h v] |
| simp only [ModuleCat.hom_zero, |
| pow_zero, one_smul, Nat.add_zero, outerCast_refl, LinearMap.id_apply] |
| erw [LinearMap.zero_apply (σ₁₂ := RingHom.id D) |
| (M₂ := D ⊗[k] term k A N (r+q)) v, zero_add] |
| | succ i => |
| erw [twoSidedParity_outerCast k D X N |
| (show (r+(q+1))+i=r+n by omega) |
| (show ((r+(q+1))+i)+1=(r+n)+1 by omega)] |
| erw [parity_outerBarSuffix k D X N ((I r).X (q+1)) (r+q) i |
| (innerTotalEmbedding k D X N r (q+1)) (innerTotalEmbedding_smul k D X N r (q+1))] |
| simp only [hδ, outerBarSuffix_neg, LinearMap.neg_apply, map_add, outerCast_smul, map_neg] |
| erw [leadingTotalEmbedding_d₁ k D X N r n i (q+1) h v] |
| erw [leadingTotalEmbedding_d₂ k D X N r n (i+1) q h v] |
| apply congrArg₂ (· + ·) (by rfl) |
| let w := outerBarSuffix k D X N ((I r).X (q+1)) (r+q) (i+1) |
| ((innerTotalEmbedding k D X N r q).comp (((I r).d (q+1) q).hom.restrictScalars k)) v |
| have hc := outerCast_trans k D X N |
| (show (r+q)+(i+1)=(r+q+1)+i by omega) |
| (show (r+q+1)+i=r+n by omega) w |
| change (-1:k)^i • -(outerCast k D X N (show (r+q+1)+i=r+n by omega) |
| (outerCast k D X N (show (r+q)+(i+1)=(r+q+1)+i by omega) w)) = |
| (-1:k)^(i+1) • outerCast k D X N (show (r+q)+(i+1)=r+n by omega) w |
| rw [hc] |
| have hs : (-1:k)^(i+1) = -((-1:k)^i) := by rw [pow_succ, mul_neg_one] |
| rw [hs] |
| exact (smul_neg _ _).trans (_root_.neg_smul _ _).symm |
|
|
| theorem leadingTotalComparison_bottom (r n i : ℕ) (h : i+0=n+1) |
| (v : (((DB).obj ((I r).X 0)).X i)) : |
| SquareZero.multibarTwoSidedDifferential k D X N (r+n) |
| (outerCast k D X N (show (r+0)+i=r+(n+1) by omega) |
| (outerBarSuffix k D X N ((I r).X 0) (r+0) i |
| (innerTotalEmbedding k D X N r 0) v)) = |
| (((K (I r)).d₁ c i 0 n ≫ leadingTotalEmbedding k D X N r n).hom v) + |
| (((K (I r)).d₂ c i 0 n ≫ leadingTotalEmbedding k D X N r n).hom v) := by |
| have hi : i=n+1 := by omega |
| subst i |
| simp only [Nat.add_zero, outerCast_refl, LinearMap.id_apply] |
| rw [leadingTotalEmbedding_d₂_zero] |
| simp only [ModuleCat.hom_zero] |
| erw [LinearMap.zero_apply (σ₁₂ := RingHom.id D) |
| (M₂ := D ⊗[k] term k A N (r+n)) v] |
| simp only [add_zero] |
| erw [leadingTotalEmbedding_d₁ k D X N r n n 0 h v] |
| simp only [Nat.add_zero, outerCast_refl, LinearMap.id_apply] |
| cases r with |
| | zero => |
| rw [innerTotalEmbedding_zero] |
| exact parity_outerBarSuffix_terminal k D X N n v |
| | succ r => |
| erw [parity_outerBarSuffix k D X N ((I (r+1)).X 0) r n |
| (innerTotalEmbedding k D X N (r+1) 0) (innerTotalEmbedding_smul k D X N (r+1) 0)] |
| have hz : (SquareZero.multibarAugmentedDifferential k D X N r).comp |
| (innerTotalEmbedding k D X N (r+1) 0) = 0 := by |
| ext w |
| exact innerTotalEmbedding_bottom k D X N r w |
| simp only [hz, outerBarSuffix_zero, LinearMap.zero_apply, map_zero, smul_zero, add_zero] |
|
|
| /-- The canonical leading total embedding intertwines its original total |
| horizontal/vertical differential with the literal square-zero-extension faces. -/ |
| theorem leadingTotalEmbedding_differential (r n : ℕ) : |
| leadingTotalEmbedding k D X N r (n+1) ≫ |
| ModuleCat.ofHom (SquareZero.multibarTwoSidedDifferential k D X N (r+n)) = |
| (barTotal k D (I r)).d (n+1) n ≫ leadingTotalEmbedding k D X N r n := by |
| apply HomologicalComplex₂.total.hom_ext |
| intro i q h |
| erw [← Category.assoc, leadingTotalEmbedding_ι, leadingTotalEmbedding_ι_d] |
| apply ModuleCat.hom_ext |
| apply LinearMap.ext |
| intro v |
| change SquareZero.multibarTwoSidedDifferential k D X N (r+n) |
| (outerCast k D X N _ (outerBarSuffix k D X N ((I r).X q) (r+q) i |
| (innerTotalEmbedding k D X N r q) v)) = _ |
| cases q with |
| | zero => exact leadingTotalComparison_bottom k D X N r n i h v |
| | succ q => exact leadingTotalComparison_positive k D X N r n i q h v |
| end DerivedBar.ParityBridge |
|
|
| end |
|
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| end OAI |
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