File size: 6,195 Bytes
216fb31 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 | 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
|