| 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 |
|
|
