| import OAI.Algebra.Finitistic.Ordinary.OrdinaryBlockTensorTerms | |
| import OAI.Algebra.Finitistic.Ordinary.OrdinaryBimoduleTensorTotal | |
| namespace OAI | |
| noncomputable section | |
| open CategoryTheory | |
| open scoped TensorProduct | |
| namespace LittleFinitistic.OrdinarySimulation.BlockTensor | |
| universe u | |
| variable (k L R : Type u) [Field k] [Ring L] [Ring R] [Algebra k L] [Algebra k R] | |
| variable (O : CochainComplex (ModuleCat.{u} (R ⊗[k] Lᵐᵒᵖ)) ℤ) | |
| variable (T : CochainComplex (ModuleCat.{u} (L ⊗[k] Rᵐᵒᵖ)) ℤ) | |
| variable (Z : CochainComplex (ModuleCat.{u} L) ℤ) | |
| variable {l : ℕ} (p : Fin l → ℤ) (hp : StrictMono p) | |
| open BimoduleComplexTensor | |
| def firstTotalEquiv (n : ℤ) : | |
| totalTerm k (L × R) (L × R) (OffDiagonal.complex k L R O T) (BlockModule.firstComplex L R Z) p n ≃ₗ[L × R] | |
| BlockModule.Second k L R (totalTerm k R L O Z p n) := | |
| (LinearEquiv.piCongrRight (fun i => firstTermEquiv k L R O T Z (p i) (n + -(p i)))).trans | |
| (show (∀ i : Fin l, BlockModule.Second k L R (term k R L O Z (p i) (n + -(p i)))) ≃ₗ[L × R] | |
| BlockModule.Second k L R (totalTerm k R L O Z p n) from LinearEquiv.refl _ _) | |
| @[simp] theorem firstTotalEquiv_apply (n : ℤ) | |
| (z : totalTerm k (L × R) (L × R) (OffDiagonal.complex k L R O T) (BlockModule.firstComplex L R Z) p n) | |
| (i : Fin l) : firstTotalEquiv k L R O T Z p n z i = | |
| firstTermEquiv k L R O T Z (p i) (n + -(p i)) (z i) := rfl | |
| theorem first_entry (n : ℤ) (i j : Fin l) : | |
| (firstTermEquiv k L R O T Z (p j) ((n+1) + -(p j))).toLinearMap.comp | |
| (entry k (L × R) (L × R) (OffDiagonal.complex k L R O T) (BlockModule.firstComplex L R Z) p n i j) = | |
| (BlockModule.secondMap k L R (entry k R L O Z p n i j)).comp | |
| (firstTermEquiv k L R O T Z (p i) (n + -(p i))).toLinearMap := by | |
| unfold entry | |
| split_ifs with h | |
| · have he : n + -(p i) = (n+1) + -(p j) := by omega | |
| have ht : (ProductModules.firstInflation L R).map (eqToHom (congrArg Z.X he)) = | |
| eqToHom (congrArg (BlockModule.firstComplex L R Z).X he) := by | |
| generalize n + -(p i) = q at he ⊢ | |
| generalize (n+1) + -(p j) = q' at he ⊢ | |
| subst q' | |
| simp only [eqToHom_refl,CategoryTheory.Functor.map_id] | |
| rfl | |
| have hh := firstTermEquiv_map k L R O T Z (O.d (p i) (p j)) (T.d (p i) (p j)) (eqToHom (congrArg Z.X he)) | |
| rw [ht,← OffDiagonal.complex_d_all] at hh | |
| exact hh | |
| · apply LinearMap.ext | |
| intro x | |
| change firstTermEquiv k L R O T Z (p j) ((n+1) + -(p j)) 0 = 0 | |
| exact (firstTermEquiv k L R O T Z (p j) ((n+1) + -(p j))).map_zero | |
| theorem first_vertical (n : ℤ) (i : Fin l) : | |
| (firstTermEquiv k L R O T Z (p i) ((n+1) + -(p i))).toLinearMap.comp | |
| (map k (L × R) (L × R) (OffDiagonal.complex k L R O T) (BlockModule.firstComplex L R Z) (𝟙 _) | |
| ((BlockModule.firstComplex L R Z).d (n + -(p i)) ((n+1) + -(p i)))) = | |
| (BlockModule.secondMap k L R (map k R L O Z (𝟙 _) (Z.d (n + -(p i)) ((n+1) + -(p i))))).comp | |
| (firstTermEquiv k L R O T Z (p i) (n + -(p i))).toLinearMap := by | |
| exact firstTermEquiv_map k L R O T Z (𝟙 (O.X (p i))) (𝟙 (T.X (p i))) (Z.d _ _) | |
| theorem first_differential (n : ℤ) | |
| (z : totalTerm k (L × R) (L × R) (OffDiagonal.complex k L R O T) (BlockModule.firstComplex L R Z) p n) : | |
| firstTotalEquiv k L R O T Z p (n+1) | |
| (differential k (L × R) (L × R) (OffDiagonal.complex k L R O T) (BlockModule.firstComplex L R Z) p n z) = | |
| differential k R L O Z p n (firstTotalEquiv k L R O T Z p n z) := by | |
| funext j | |
| change firstTermEquiv k L R O T Z (p j) ((n+1) + -(p j)) | |
| (differential k (L × R) (L × R) (OffDiagonal.complex k L R O T) (BlockModule.firstComplex L R Z) p n z j) = | |
| differential k R L O Z p n (firstTotalEquiv k L R O T Z p n z) j | |
| simp only [differential, | |
| LinearMap.pi_apply,LinearMap.add_apply,LinearMap.comp_apply,LinearMap.proj_apply, | |
| LinearMap.sum_apply,map_add,map_sum] | |
| apply congrArg₂ (· + ·) | |
| · change firstTermEquiv k L R O T Z (p j) ((n+1) + -(p j)) | |
| ((((-(p j)).negOnePow : ℤ)) • | |
| (map k (L × R) (L × R) (OffDiagonal.complex k L R O T) (BlockModule.firstComplex L R Z) (𝟙 _) | |
| ((BlockModule.firstComplex L R Z).d (n + -(p j)) ((n+1) + -(p j))) (z j))) = | |
| (((-(p j)).negOnePow : ℤ)) • | |
| (map k R L O Z (𝟙 _) (Z.d (n + -(p j)) ((n+1) + -(p j)))) | |
| (firstTermEquiv k L R O T Z (p j) (n + -(p j)) (z j)) | |
| rw [map_zsmul] | |
| congr 1 | |
| exact congrArg (fun f => f (z j)) (first_vertical k L R O T Z p n j) | |
| · erw [LinearMap.sum_apply] | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| exact congrArg (fun f => f (z i)) (first_entry k L R O T Z p n i j) | |
| /-- Actual tensor with the off-diagonal complex on a first-block input is the second-block O-total. -/ | |
| def firstTotalIso : | |
| total k (L × R) (L × R) (OffDiagonal.complex k L R O T) (BlockModule.firstComplex L R Z) p hp ≅ | |
| BlockModule.secondComplex L R (total k R L O Z p hp) := | |
| HomologicalComplex.Hom.isoOfComponents (fun n => (firstTotalEquiv k L R O T Z p n).toModuleIso) (by | |
| intro n m h | |
| change n+1=m at h | |
| subst m | |
| have h₁ : (total k (L × R) (L × R) (OffDiagonal.complex k L R O T) (BlockModule.firstComplex L R Z) p hp).d n (n+1) = | |
| ModuleCat.ofHom (differential k (L × R) (L × R) (OffDiagonal.complex k L R O T) (BlockModule.firstComplex L R Z) p n) := | |
| CochainComplex.of_d (fun n => ModuleCat.of (L × R) (totalTerm k (L × R) (L × R) (OffDiagonal.complex k L R O T) (BlockModule.firstComplex L R Z) p n)) | |
| (fun n => ModuleCat.ofHom (differential k (L × R) (L × R) (OffDiagonal.complex k L R O T) (BlockModule.firstComplex L R Z) p n)) n | |
| have h₂ : (total k R L O Z p hp).d n (n+1) = ModuleCat.ofHom (differential k R L O Z p n) := | |
| CochainComplex.of_d (fun n => ModuleCat.of R (totalTerm k R L O Z p n)) | |
| (fun n => ModuleCat.ofHom (differential k R L O Z p n)) n | |
| change _ ≫ (ProductModules.secondInflation L R).map ((total k R L O Z p hp).d n (n+1)) = _ | |
| rw [h₁,h₂] | |
| apply ModuleCat.hom_ext | |
| apply LinearMap.ext | |
| intro z | |
| exact (first_differential k L R O T Z p n z).symm) | |
| end LittleFinitistic.OrdinarySimulation.BlockTensor | |
| end | |
| end OAI | |