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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 | import Mathlib
import OAI.Algebra.FinitisticAsymmetry.Transport.ScalarRestriction
namespace OAI
section
section
noncomputable section
open CategoryTheory
namespace LittleFinitistic
universe u
variable (A : Type u) [Ring A]
namespace TriangularConstruction
universe v
variable (C E U : Type v)
def OffDiagonal (_C _E U : Type v) := U
def offValue {C E U : Type v} (x : OffDiagonal C E U) : U := x
namespace OffDiagonal
variable [Ring C] [Ring E] [AddCommGroup U]
instance : AddCommGroup (OffDiagonal C E U) := inferInstanceAs (AddCommGroup U)
instance [Module E U] : Module (C × E) (OffDiagonal C E U) :=
Module.compHom U (RingHom.snd C E)
instance [Module Cᵐᵒᵖ U] : Module (C × E)ᵐᵒᵖ (OffDiagonal C E U) :=
Module.compHom U (RingHom.fst C E).op
instance [Module E U] [Module Cᵐᵒᵖ U] [SMulCommClass E Cᵐᵒᵖ U] :
SMulCommClass (C × E) (C × E)ᵐᵒᵖ (OffDiagonal C E U) where
smul_comm p q x := by
change p.2 • (MulOpposite.op q.unop.1 • (offValue x)) =
MulOpposite.op q.unop.1 • (p.2 • (offValue x))
exact smul_comm _ _ _
variable (k : Type v) [Field k] [_root_.Algebra k C] [_root_.Algebra k E] [Module k U]
instance : Module k (OffDiagonal C E U) := inferInstanceAs (Module k U)
instance [Module E U] [IsScalarTower k E U] :
IsScalarTower k (C × E) (OffDiagonal C E U) where
smul_assoc r p x := by
change (r • p.2) • (offValue x) = r • (p.2 • (offValue x))
exact smul_assoc _ _ _
instance [Module Cᵐᵒᵖ U] [IsScalarTower k Cᵐᵒᵖ U] :
IsScalarTower k (C × E)ᵐᵒᵖ (OffDiagonal C E U) where
smul_assoc r p x := by
change (r • MulOpposite.op p.unop.1) • (offValue x) =
r • (MulOpposite.op p.unop.1 • (offValue x))
exact smul_assoc _ _ _
instance [FiniteDimensional k U] : FiniteDimensional k (OffDiagonal C E U) :=
inferInstanceAs (FiniteDimensional k U)
end OffDiagonal
abbrev Algebra := TrivSqZeroExt (C × E) (OffDiagonal C E U)
variable [Ring C] [Ring E] [AddCommGroup U] [Module E U] [Module Cᵐᵒᵖ U]
[SMulCommClass E Cᵐᵒᵖ U]
omit [SMulCommClass E Cᵐᵒᵖ U] in
lemma mul_formula (x y : Algebra C E U) :
x * y = ((x.fst.1 * y.fst.1, x.fst.2 * y.fst.2),
x.fst.2 • (offValue y.snd) + MulOpposite.op y.fst.1 • (offValue x.snd)) := rfl
def toC : Algebra C E U →+* C where
toFun x := x.fst.1
map_zero' := rfl
map_one' := rfl
map_add' _ _ := rfl
map_mul' _ _ := rfl
omit [SMulCommClass E Cᵐᵒᵖ U] in
lemma toC_surjective : Function.Surjective (toC C E U) :=
fun c => ⟨((c, 0), 0), rfl⟩
def rightCornerSection :
((ModuleCat.restrictScalars (toC C E U).op).obj (ModuleCat.of Cᵐᵒᵖ Cᵐᵒᵖ)) →ₗ[
(Algebra C E U)ᵐᵒᵖ] (Algebra C E U)ᵐᵒᵖ where
toFun c := MulOpposite.op ((c.unop, 0), 0)
map_add' c d := by
apply MulOpposite.unop_injective
apply TrivSqZeroExt.ext
· apply Prod.ext
· rfl
· exact (zero_add (0 : E)).symm
· exact (zero_add (0 : U)).symm
map_smul' r c := by
apply MulOpposite.unop_injective
apply TrivSqZeroExt.ext
· apply Prod.ext
· rfl
· exact (zero_mul _).symm
· change 0 = (0 : E) • (offValue r.unop.snd) + MulOpposite.op r.unop.fst.1 • (0 : U)
simp
lemma rightCorner_projective :
Module.Projective (Algebra C E U)ᵐᵒᵖ
((ModuleCat.restrictScalars (toC C E U).op).obj (ModuleCat.of Cᵐᵒᵖ Cᵐᵒᵖ)) := by
apply Module.Projective.of_split (rightCornerSection C E U)
(Module.compHom.toLinearMap (toC C E U).op)
ext x
rfl
lemma right_little_lower_bound :
littleFinitisticDimension Cᵐᵒᵖ ≤ littleFinitisticDimension (Algebra C E U)ᵐᵒᵖ := by
let := rightCorner_projective C E U
apply littleFinitisticDimension_le_restrictScalars (toC C E U).op
intro c
obtain ⟨t, ht⟩ := toC_surjective C E U c.unop
exact ⟨MulOpposite.op t, congrArg MulOpposite.op ht⟩
lemma right_big_lower_bound :
bigFinitisticDimension Cᵐᵒᵖ ≤ bigFinitisticDimension (Algebra C E U)ᵐᵒᵖ := by
let := rightCorner_projective C E U
apply bigFinitisticDimension_le_restrictScalars (toC C E U).op
intro c
obtain ⟨t, ht⟩ := toC_surjective C E U c.unop
exact ⟨MulOpposite.op t, congrArg MulOpposite.op ht⟩
variable (k : Type v) [Field k] [_root_.Algebra k C] [_root_.Algebra k E] [Module k U]
[IsScalarTower k E U] [IsScalarTower k Cᵐᵒᵖ U]
[FiniteDimensional k C] [FiniteDimensional k E] [FiniteDimensional k U]
omit [Module E U] [Module Cᵐᵒᵖ U] [SMulCommClass E Cᵐᵒᵖ U]
[IsScalarTower k E U] [IsScalarTower k Cᵐᵒᵖ U] in
lemma finiteDimensional : FiniteDimensional k (Algebra C E U) :=
inferInstanceAs (FiniteDimensional k ((C × E) × OffDiagonal C E U))
end TriangularConstruction
end LittleFinitistic
end
end
end
end OAI
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