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ae00908 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 | import OAI.Algebra.Finitistic.Rectification.ColumnProperties
namespace OAI
open CategoryTheory CategoryTheory.Limits HomologicalComplex
open scoped TensorProduct ModuleCat.Algebra
noncomputable section
namespace ExternalTensor.Section4.ArrowVertex
attribute [local instance] ExternalTensor.bimodule ExternalTensor.rightModule
universe u
variable (k B : Type u) [Field k] [Ring B] [Algebra k B]
/-- Left multiplication on a bimodule, regarded as a right-linear map. -/
def left (a : B) (N : ModuleCat.{u} (Enveloping k B)) :
(forgetRight k B).obj N →ₗ[Bᵐᵒᵖ] (forgetRight k B).obj N where
toFun x := (a ⊗ₜ[k] (1 : Bᵐᵒᵖ)) • (show N from x)
map_add' x y := smul_add (a ⊗ₜ[k] (1 : Bᵐᵒᵖ)) (show N from x) (show N from y)
map_smul' c x := by
change (a ⊗ₜ[k] (1 : Bᵐᵒᵖ)) • ((1 ⊗ₜ[k] c : Enveloping k B) • (show N from x)) =
(1 ⊗ₜ[k] c : Enveloping k B) • ((a ⊗ₜ[k] (1 : Bᵐᵒᵖ)) • (show N from x))
erw [← mul_smul, ← mul_smul]
simp only [Algebra.TensorProduct.tmul_mul_tmul, mul_one, one_mul]
@[simp] theorem left_mul (a b : B) (N : ModuleCat.{u} (Enveloping k B))
(x : (forgetRight k B).obj N) : left k B (a * b) N x = left k B a N (left k B b N x) := by
change ((a * b) ⊗ₜ[k] (1 : Bᵐᵒᵖ)) • (show N from x) =
(a ⊗ₜ[k] (1 : Bᵐᵒᵖ)) • ((b ⊗ₜ[k] (1 : Bᵐᵒᵖ)) • (show N from x))
erw [← mul_smul]
simp only [Algebra.TensorProduct.tmul_mul_tmul, mul_one]
@[simp] theorem map_left {N N' : ModuleCat.{u} (Enveloping k B)}
(f : N ⟶ N') (a : B) (x : (forgetRight k B).obj N) :
(forgetRight k B).map f (left k B a N x) =
left k B a N' ((forgetRight k B).map f x) := f.hom.map_smul _ _
/-- The actual fixed submodule eN, for an idempotent e. -/
def space (e : B) (N : ModuleCat.{u} (Enveloping k B)) :
Submodule Bᵐᵒᵖ ((forgetRight k B).obj N) :=
LinearMap.ker (left k B e N - LinearMap.id)
@[simp] theorem mem_space (e : B) (N : ModuleCat.{u} (Enveloping k B))
(x : (forgetRight k B).obj N) : x ∈ space k B e N ↔ left k B e N x = x := by
simp [space, LinearMap.mem_ker, sub_eq_zero]
/-- For idempotents the fixed submodule is exactly the image eN. -/
theorem space_eq_range (e : B) (he : e * e = e)
(N : ModuleCat.{u} (Enveloping k B)) :
space k B e N = (left k B e N).range := by
ext x
rw [mem_space, LinearMap.mem_range]
constructor
· intro hx
exact ⟨x, hx⟩
· rintro ⟨y, rfl⟩
rw [← left_mul, he]
/-- Vertex restriction retains the original elements, rather than chosen coordinates. -/
def vertex (e : B) : ModuleCat.{u} (Enveloping k B) ⥤ ModuleCat.{u} Bᵐᵒᵖ where
obj N := ModuleCat.of _ (space k B e N)
map f := ModuleCat.ofHom
{ toFun x := ⟨(forgetRight k B).map f x.val, by
rw [mem_space, ← map_left, (mem_space k B e _ _).1 x.property]⟩
map_add' x y := Subtype.ext (map_add _ _ _)
map_smul' c x := Subtype.ext (map_smul _ _ _) }
map_id N := by ext x; rfl
map_comp f g := by ext x; rfl
instance vertex_additive (e : B) : (vertex k B e).Additive where
map_add := by intros; ext x; rfl
abbrev vertexComplex (e : B) : BiComplex k B ⥤ RightComplex B :=
(vertex k B e).mapHomologicalComplex (ComplexShape.up ℤ)
/-- The intervertex map is precisely left multiplication by a. -/
def arrow (ei ej a : B) (hja : ej * a = a) : vertex k B ei ⟶ vertex k B ej where
app N := ModuleCat.ofHom
{ toFun x := ⟨left k B a N x.val, by
rw [mem_space, ← left_mul, hja]⟩
map_add' x y := Subtype.ext (map_add _ _ _)
map_smul' c x := Subtype.ext (map_smul _ _ _) }
naturality {N N'} f := by ext x; apply Subtype.ext; exact (map_left k B f a x.val).symm
abbrev arrowComplex (ei ej a : B) (hja : ej * a = a) (P : BiComplex k B) :
(vertexComplex k B ei).obj P ⟶ (vertexComplex k B ej).obj P :=
((arrow k B ei ej a hja).mapHomologicalComplex (ComplexShape.up ℤ)).app P
@[simp] theorem arrow_original (ei ej a : B) (hja : ej * a = a)
(P : BiComplex k B) (n : ℤ) (x : ((vertexComplex k B ei).obj P).X n) :
(((arrowComplex k B ei ej a hja P).f n) x).val =
(a ⊗ₜ[k] (1 : Bᵐᵒᵖ)) • (show P.X n from x.val) := rfl
/-- Composition uses multiplication in B in the original arrow order. -/
@[reassoc] theorem arrow_comp (ei ej el a b : B) (hja : ej * a = a)
(hlb : el * b = b) (hlba : el * (b * a) = b * a) (P : BiComplex k B) :
arrowComplex k B ei ej a hja P ≫ arrowComplex k B ej el b hlb P =
arrowComplex k B ei el (b * a) hlba P := by
ext n x
apply Subtype.ext
exact (left_mul k B b a (P.X n) x.val).symm
/-- The trivial path acts as the identity on its vertex. -/
@[simp] theorem arrow_id (e : B) (he : e * e = e) (P : BiComplex k B) :
arrowComplex k B e e e he P = 𝟙 ((vertexComplex k B e).obj P) := by
ext n x
apply Subtype.ext
exact (mem_space k B e (P.X n) x.val).1 x.property
/-- The original diagonal tensor e tensor n in its actual fixed submodule. -/
def diagonal (e : B) (he : e * e = e) :
𝟭 (ModuleCat.{u} Bᵐᵒᵖ) ⟶ columnFunctor k B e ⋙ vertex k B e where
app N := ModuleCat.ofHom
{ toFun n := ⟨generator B e ⊗ₜ[k] n, by
erw [mem_space]
change (e • generator B e) ⊗ₜ[k] ((1 : Bᵐᵒᵖ) • n) = generator B e ⊗ₜ[k] n
rw [one_smul]
congr 1
exact Subtype.ext he⟩
map_add' x y := Subtype.ext (TensorProduct.tmul_add _ _ _)
map_smul' c x := by
apply Subtype.ext
change generator B e ⊗ₜ[k] (c • x) =
(1 ⊗ₜ[k] c : Enveloping k B) • (generator B e ⊗ₜ[k] x)
rw [ExternalTensor.smul_tmul, one_smul] }
naturality {N N'} f := by ext x; rfl
/-- Original insertion into a single induced column, degree by degree. -/
def diagonalComplex (e : B) (he : e * e = e) (D : RightComplex B) :
D ⟶ (vertexComplex k B e).obj ((columnComplex k B e).obj D) where
f n := (diagonal k B e he).app (D.X n)
comm' i j _hij := (diagonal k B e he).naturality (D.d i j)
@[reassoc] theorem diagonal_naturality (e : B) (he : e * e = e)
{D E : RightComplex B} (f : D ⟶ E) :
f ≫ diagonalComplex k B e he E = diagonalComplex k B e he D ≫
(vertexComplex k B e).map ((columnComplex k B e).map f) := by
ext n x
rfl
/-- Right multiplication in p at e_j agrees with left arrow action at e_i. -/
@[reassoc] theorem diagonal_path (ei ej a : B) (hei : ei * ei = ei)
(hej : ej * ej = ej) (hai : a * ei = a) (hja : ej * a = a)
(D : RightComplex B) :
diagonalComplex k B ej hej D ≫ (vertexComplex k B ej).map (path k B ei ej a hai D) =
diagonalComplex k B ei hei D ≫
arrowComplex k B ei ej a hja ((columnComplex k B ei).obj D) := by
ext n x
apply Subtype.ext
change (pathMap ei ej a hai (generator B ej)) ⊗ₜ[k] x =
(a • generator B ei) ⊗ₜ[k] ((1 : Bᵐᵒᵖ) • x)
rw [one_smul]
congr 1
apply Subtype.ext
exact hja.trans hai.symm
@[reassoc] theorem arrow_naturality (ei ej a : B) (hja : ej * a = a)
{P Q : BiComplex k B} (f : P ⟶ Q) :
(vertexComplex k B ei).map f ≫ arrowComplex k B ei ej a hja Q =
arrowComplex k B ei ej a hja P ≫ (vertexComplex k B ej).map f :=
((arrow k B ei ej a hja).mapHomologicalComplex (ComplexShape.up ℤ)).naturality f
end ExternalTensor.Section4.ArrowVertex
end
end OAI
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