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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 | import OAI.Algebra.Finitistic.Encoding.DirectedBound
import Mathlib.LinearAlgebra.FiniteDimensional.Basic
import Mathlib.Tactic.FinCases
namespace OAI
noncomputable section
namespace LittleFinitistic
universe u v
variable {k V W U : Type u} [Field k]
[AddCommGroup V] [Module k V] [AddCommGroup W] [Module k W]
[AddCommGroup U] [Module k U]
/-- Coordinates by the six possible path endpoints: three vertices,
0-to-1 arrows, 1-to-2 arrows, and length-two paths modulo relations. -/
abbrev ThreeVertexAlgebra (_μ : W →ₗ[k] V →ₗ[k] U) := k × k × k × V × W × U
namespace ThreeVertexAlgebra
variable (μ : W →ₗ[k] V →ₗ[k] U)
instance : AddCommGroup (ThreeVertexAlgebra μ) :=
inferInstanceAs (AddCommGroup (k × k × k × V × W × U))
instance : Module k (ThreeVertexAlgebra μ) :=
inferInstanceAs (Module k (k × k × k × V × W × U))
@[simp] theorem zero_eq : (0 : ThreeVertexAlgebra μ) = (0, 0, 0, 0, 0, 0) := rfl
@[simp] theorem add_eq (x y : ThreeVertexAlgebra μ) :
x + y = (x.1+y.1, x.2.1+y.2.1, x.2.2.1+y.2.2.1,
x.2.2.2.1+y.2.2.2.1, x.2.2.2.2.1+y.2.2.2.2.1,
x.2.2.2.2.2+y.2.2.2.2.2) := rfl
@[simp] theorem smul_eq (c : k) (x : ThreeVertexAlgebra μ) :
c • x = (c*x.1, c*x.2.1, c*x.2.2.1,
c • x.2.2.2.1, c • x.2.2.2.2.1, c • x.2.2.2.2.2) := rfl
def mul (x y : ThreeVertexAlgebra μ) : ThreeVertexAlgebra μ :=
(x.1 * y.1, x.2.1 * y.2.1, x.2.2.1 * y.2.2.1,
y.1 • x.2.2.2.1 + x.2.1 • y.2.2.2.1,
y.2.1 • x.2.2.2.2.1 + x.2.2.1 • y.2.2.2.2.1,
y.1 • x.2.2.2.2.2 + μ x.2.2.2.2.1 y.2.2.2.1 + x.2.2.1 • y.2.2.2.2.2)
def unit : ThreeVertexAlgebra μ := (1, 1, 1, 0, 0, 0)
@[ext] theorem ext (x y : ThreeVertexAlgebra μ)
(h0 : x.1 = y.1) (h1 : x.2.1 = y.2.1) (h2 : x.2.2.1 = y.2.2.1)
(hv : x.2.2.2.1 = y.2.2.2.1) (hw : x.2.2.2.2.1 = y.2.2.2.2.1)
(hu : x.2.2.2.2.2 = y.2.2.2.2.2) : x = y :=
Prod.ext h0 (Prod.ext h1 (Prod.ext h2 (Prod.ext hv (Prod.ext hw hu))))
instance : Ring (ThreeVertexAlgebra μ) where
__ := (inferInstance : AddCommGroup (ThreeVertexAlgebra μ))
mul := mul μ
one := unit μ
mul_assoc x y z := by
change mul μ (mul μ x y) z = mul μ x (mul μ y z)
apply Prod.ext <;> dsimp [mul]
· ring
apply Prod.ext
· ring
apply Prod.ext
· ring
apply Prod.ext
· simp only [smul_add, smul_smul]; module
apply Prod.ext
· simp only [smul_add, smul_smul]; module
· simp only [map_add, map_smul, LinearMap.add_apply, LinearMap.smul_apply,
smul_add, smul_smul]; module
one_mul x := by
change mul μ (unit μ) x = x
apply Prod.ext <;> dsimp [mul, unit]
· simp
apply Prod.ext
· simp
apply Prod.ext
· simp
apply Prod.ext
· simp
apply Prod.ext <;> simp
mul_one x := by
change mul μ x (unit μ) = x
apply Prod.ext <;> dsimp [mul, unit]
· simp
apply Prod.ext
· simp
apply Prod.ext
· simp
apply Prod.ext
· simp
apply Prod.ext <;> simp
left_distrib x y z := by
change mul μ x (y + z) = mul μ x y + mul μ x z
rw [add_eq μ (mul μ x y) (mul μ x z), add_eq μ y z]
ext <;> simp [mul, mul_add, smul_add, add_smul, map_add] <;> abel
right_distrib x y z := by
change mul μ (x + y) z = mul μ x z + mul μ y z
rw [add_eq μ (mul μ x z) (mul μ y z), add_eq μ x y]
ext <;> simp [mul, add_mul, smul_add, add_smul, map_add] <;> abel
zero_mul x := by
change (mul μ 0 x : k × k × k × V × W × U) = 0
ext <;> simp [mul]
mul_zero x := by
change (mul μ x 0 : k × k × k × V × W × U) = 0
ext <;> simp [mul]
@[simp] theorem mul_eq (x y : ThreeVertexAlgebra μ) : x * y = mul μ x y := rfl
@[simp] theorem one_eq : (1 : ThreeVertexAlgebra μ) = unit μ := rfl
instance : Algebra k (ThreeVertexAlgebra μ) where
algebraMap :=
{ toFun := fun c => (c, c, c, 0, 0, 0)
map_zero' := (zero_eq μ).symm
map_one' := rfl
map_add' := fun a b => by
change ((a+b, a+b, a+b, 0, 0, 0) : ThreeVertexAlgebra μ) =
(a, a, a, 0, 0, 0) + (b, b, b, 0, 0, 0)
rw [add_eq μ]
simp
map_mul' := fun a b => by
change ((a*b, a*b, a*b, 0, 0, 0) : ThreeVertexAlgebra μ) =
mul μ (a, a, a, 0, 0, 0) (b, b, b, 0, 0, 0)
simp [mul] }
commutes' c x := by
change mul μ (c, c, c, 0, 0, 0) x = mul μ x (c, c, c, 0, 0, 0)
ext <;> simp [mul, mul_comm]
smul_def' c x := by
change c • x = mul μ (c, c, c, 0, 0, 0) x
rw [smul_eq μ]
simp [mul]
instance [FiniteDimensional k V] [FiniteDimensional k W] [FiniteDimensional k U] :
FiniteDimensional k (ThreeVertexAlgebra μ) :=
inferInstanceAs (FiniteDimensional k (k × k × k × V × W × U))
/-- The three vertex idempotents, in the manuscript's order. -/
def vertex : Fin 3 → ThreeVertexAlgebra μ
| ⟨0, _⟩ => (1, 0, 0, 0, 0, 0)
| ⟨1, _⟩ => (0, 1, 0, 0, 0, 0)
| ⟨2, _⟩ => (0, 0, 1, 0, 0, 0)
def directedSystem : DirectedBound.DirectedSystem k (ThreeVertexAlgebra μ) 3 where
e := vertex μ
idem i := by fin_cases i <;> ext <;> simp [vertex, mul]
orthogonal i j hij := by
fin_cases i <;> fin_cases j <;> simp_all [vertex, mul]
complete := by
simp [Fin.sum_univ_succ, vertex, unit, add_eq μ]
diagonal i a := by
fin_cases i
· exact ⟨a.1, by ext <;> simp [vertex, mul, smul_eq μ]⟩
· exact ⟨a.2.1, by ext <;> simp [vertex, mul, smul_eq μ]⟩
· exact ⟨a.2.2.1, by ext <;> simp [vertex, mul, smul_eq μ]⟩
directed i j hij a := by
fin_cases i <;> fin_cases j <;> simp_all [vertex, mul]
open CategoryTheory
/-- The three-vertex coordinate algebra has global dimension at most two on both sides. -/
theorem globalDimension_le_two :
(⨆ N : ModuleCat.{max u v} (ThreeVertexAlgebra μ), projectiveDimension N) ≤ (2 : ℕ) ∧
(⨆ N : ModuleCat.{max u v} (ThreeVertexAlgebra μ)ᵐᵒᵖ, projectiveDimension N) ≤ (2 : ℕ) :=
(directedSystem μ).both_sides_globalDimension_le
end ThreeVertexAlgebra
end LittleFinitistic
end
end OAI
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