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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 | import OAI.Algebra.Finitistic.Ordinary.OrdinaryChainAlgebra
import OAI.Algebra.Finitistic.Tensor.IdempotentTensor
import Mathlib.Algebra.Module.Projective
namespace OAI
noncomputable section
namespace LittleFinitistic.OrdinarySimulation
universe u
variable {k : Type u} [Field k] {l : ℕ}
namespace ChainAlgebra
@[simp] theorem arrow_mul (i : Fin l) (x : ChainAlgebra k l) :
arrow k l i * x = x.1 i.castSucc • arrow k l i := by
apply Prod.ext <;> funext j
· change (0:k) * x.1 j = x.1 i.castSucc * 0
simp
· change 0 * x.2 j + (Pi.single i 1 : Fin l → k) j * x.1 j.castSucc =
x.1 i.castSucc * (Pi.single i 1 : Fin l → k) j
by_cases h : j = i <;> simp [h]
@[simp] theorem smul_diagonal (c : k) (x : ChainAlgebra k l) (i : Fin (l+1)) :
(c • x).1 i = c * x.1 i := rfl
@[simp] theorem smul_radical (c : k) (x : ChainAlgebra k l) (i : Fin l) :
(c • x).2 i = c * x.2 i := rfl
@[simp] theorem vertex_diagonal (i j : Fin (l+1)) :
(vertex k l i).1 j = (Pi.single i 1 : Fin (l+1) → k) j := rfl
@[simp] theorem arrow_diagonal (i : Fin l) (j : Fin (l+1)) :
(arrow k l i).1 j = 0 := rfl
@[simp] theorem arrow_radical (i j : Fin l) :
(arrow k l i).2 j = (Pi.single i 1 : Fin l → k) j := rfl
/-- The two displayed coordinates of each noninitial right projective. -/
theorem vertex_succ_mul (i : Fin l) (x : ChainAlgebra k l) :
vertex k l i.succ * x = x.1 i.succ • vertex k l i.succ + x.2 i • arrow k l i := by
rw [add_eq, smul_eq, smul_eq]
apply Prod.ext <;> funext j
· change (Pi.single i.succ 1 : Fin (l+1) → k) j * x.1 j =
x.1 i.succ * (Pi.single i.succ 1 : Fin (l+1) → k) j + x.2 i * 0
by_cases h : j = i.succ <;> simp [h]
· change (Pi.single i.succ 1 : Fin (l+1) → k) j.succ * x.2 j + 0 * x.1 j.castSucc =
x.1 i.succ * 0 + x.2 i * (Pi.single i 1 : Fin l → k) j
by_cases h : j = i
· subst j; simp
· have hs : j.succ ≠ i.succ := fun e => h (Fin.succ_injective _ e)
simp [h, hs]
/-- The initial projective has no radical coordinate. -/
theorem vertex_zero_mul (x : ChainAlgebra k l) :
vertex k l 0 * x = x.1 0 • vertex k l 0 := by
rw [smul_eq]
apply Prod.ext <;> funext j
· change (Pi.single 0 1 : Fin (l+1) → k) j * x.1 j =
x.1 0 * (Pi.single 0 1 : Fin (l+1) → k) j
by_cases h : j = 0 <;> simp [h]
· change (Pi.single 0 1 : Fin (l+1) → k) j.succ * x.2 j + 0 * x.1 j.castSucc = x.1 0 * 0
simp
/-- Actual right projectives in the source resolution. -/
abbrev rightTerm (i : Fin (l+1)) := IdempotentTensor.rightIdeal (vertex k l i)
def leftMultiply (x : ChainAlgebra k l) : ChainAlgebra k l →ₗ[(ChainAlgebra k l)ᵐᵒᵖ] ChainAlgebra k l where
toFun y := x*y
map_add' := mul_add x
map_smul' r y := (mul_assoc x y r.unop).symm
def retract (i : Fin (l+1)) : ChainAlgebra k l →ₗ[(ChainAlgebra k l)ᵐᵒᵖ] rightTerm (k := k) i :=
(leftMultiply (vertex k l i)).codRestrict _ (fun x => by
change vertex k l i * (vertex k l i * x) = vertex k l i * x
rw [← mul_assoc, vertex_idempotent])
def rightRegularEquiv : (ChainAlgebra k l)ᵐᵒᵖ ≃ₗ[(ChainAlgebra k l)ᵐᵒᵖ] ChainAlgebra k l where
toFun := MulOpposite.unop
invFun := MulOpposite.op
left_inv _ := rfl
right_inv _ := rfl
map_add' _ _ := rfl
map_smul' _ _ := rfl
theorem rightTerm_projective (i : Fin (l+1)) :
Module.Projective (ChainAlgebra k l)ᵐᵒᵖ (rightTerm (k := k) i) := by
let : Module.Projective (ChainAlgebra k l)ᵐᵒᵖ (ChainAlgebra k l) := Module.Projective.of_equiv rightRegularEquiv
apply Module.Projective.of_split (rightTerm (k := k) i).subtype (retract i)
ext x
exact x.property
/-- The differential is actual left multiplication by the source arrow. -/
def resolutionMap (i : Fin l) : rightTerm (k := k) i.castSucc →ₗ[(ChainAlgebra k l)ᵐᵒᵖ] rightTerm (k := k) i.succ :=
((leftMultiply (arrow k l i)).comp (rightTerm (k := k) i.castSucc).subtype).codRestrict _ (fun x => by
change vertex k l i.succ * (arrow k l i * x.val) = arrow k l i * x.val
rw [← mul_assoc, vertex_target_arrow])
@[simp] theorem resolutionMap_value (i : Fin l) (x : rightTerm (k := k) i.castSucc) :
(resolutionMap (k := k) i x).val = x.val.1 i.castSucc • arrow k l i := arrow_mul i x.val
/-- This is the explicit two-coordinate expansion, not a dimension argument. -/
theorem rightTerm_succ_expansion (i : Fin l) (x : rightTerm (k := k) i.succ) :
x.val = x.val.1 i.succ • vertex k l i.succ + x.val.2 i • arrow k l i := by
rw [← vertex_succ_mul]
exact x.property.symm
/-- The source's resolution differentials square to zero. -/
theorem resolutionMap_square (i : ℕ) (hi : i+1 < l) :
(resolutionMap (k := k) (⟨i+1,hi⟩ : Fin l)).comp
(resolutionMap (k := k) (⟨i,by omega⟩ : Fin l)) = 0 := by
ext x
change arrow k l ⟨i+1,hi⟩ * (arrow k l ⟨i,by omega⟩ * x.val) = 0
rw [← mul_assoc, arrow_mul_arrow, zero_mul]
theorem resolutionMap_zero_iff (i : Fin l) (x : rightTerm (k := k) i.castSucc) :
resolutionMap (k := k) i x = 0 ↔ x.val.1 i.castSucc = 0 := by
constructor
· intro h
have hr := congrArg (fun y : rightTerm (k := k) i.succ => y.val.2 i) h
rw [resolutionMap_value, smul_radical, arrow_radical] at hr
simpa using hr
· intro h
apply Subtype.ext
rw [resolutionMap_value, h, zero_smul]
rfl
/-- Exact description of the image: the one-dimensional radical in the next term. -/
theorem resolutionMap_range (i : Fin l) (x : rightTerm (k := k) i.succ) :
(∃ y, resolutionMap (k := k) i y = x) ↔ x.val.1 i.succ = 0 := by
constructor
· rintro ⟨y,rfl⟩
rw [resolutionMap_value, smul_diagonal, arrow_diagonal, mul_zero]
· intro hx
let y : rightTerm (k := k) i.castSucc := ⟨x.val.2 i • vertex k l i.castSucc, by
change vertex k l i.castSucc * (x.val.2 i • vertex k l i.castSucc) =
x.val.2 i • vertex k l i.castSucc
rw [mul_smul_comm, vertex_idempotent]⟩
refine ⟨y, ?_⟩
apply Subtype.ext
rw [resolutionMap_value]
have hy : y.val.1 i.castSucc = x.val.2 i := by
dsimp [y]
simp
rw [hy]
have he := rightTerm_succ_expansion i x
rw [hx, zero_smul, zero_add] at he
exact he.symm
/-- Exactness at every interior projective, with no assumed resolution predicate. -/
theorem resolution_exact (i : ℕ) (hi : i+1 < l)
(x : rightTerm (k := k) (⟨i+1,by omega⟩ : Fin (l+1))) :
resolutionMap (k := k) (⟨i+1,hi⟩ : Fin l) x = 0 ↔
∃ y : rightTerm (k := k) (⟨i,by omega⟩ : Fin (l+1)), resolutionMap (k := k) (⟨i,by omega⟩ : Fin l) y = x := by
exact (resolutionMap_zero_iff (⟨i+1,hi⟩ : Fin l) x).trans
(resolutionMap_range (⟨i,by omega⟩ : Fin l) x).symm
/-- Injectivity of the first nonzero differential in the resolution. -/
theorem resolution_first_injective (hl : 0 < l) :
Function.Injective (resolutionMap (k := k) (⟨0,hl⟩ : Fin l)) := by
intro x y hxy
have hx : resolutionMap (k := k) (⟨0,hl⟩ : Fin l) (x-y) = 0 := by
rw [map_sub, hxy, sub_self]
have hz := (resolutionMap_zero_iff _ (x-y)).mp hx
have he := vertex_zero_mul (x-y).val
have hp : vertex k l 0 * (x-y).val = (x-y).val := (x-y).property
change (x-y).val.1 0 = 0 at hz
rw [hp, hz, zero_smul] at he
have hsub : x-y = 0 := Subtype.ext he
exact sub_eq_zero.mp hsub
end ChainAlgebra
end LittleFinitistic.OrdinarySimulation
end
end OAI
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