| import Mathlib |
| import OAI.Algebra.FinitisticAsymmetry.Triangular.Algebra |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open CategoryTheory |
| namespace LittleFinitistic |
| universe u |
| variable (A : Type u) [Ring A] |
| namespace CummingsConstruction |
| variable {k C ι : Type u} [Field k] [Ring C] [Algebra k C] |
|
|
| abbrev NewCorner (k ι : Type u) := ι → DualNumber k |
| |
| instance [Finite ι] : FiniteDimensional k (NewCorner k ι) := by |
| let : FiniteDimensional k (DualNumber k) := inferInstanceAs (FiniteDimensional k (k × k)) |
| exact inferInstanceAs (FiniteDimensional k (ι → DualNumber k)) |
|
|
| def newAugmentation : NewCorner k ι →ₐ[k] (ι → k) := |
| AlgHom.pi fun i => (TrivSqZeroExt.fstHom k k k).comp (Pi.evalAlgHom k (fun _ : ι => DualNumber k) i) |
|
|
| def SimpleSum (_χ : C →ₐ[k] (ι → k)) := ι → k |
|
|
| namespace SimpleSum |
| def value {χ : C →ₐ[k] (ι → k)} (x : SimpleSum χ) : ι → k := x |
| variable (χ : C →ₐ[k] (ι → k)) |
| instance : AddCommGroup (SimpleSum χ) := inferInstanceAs (AddCommGroup (ι → k)) |
| instance : Module k (SimpleSum χ) := inferInstanceAs (Module k (ι → k)) |
| instance [Finite ι] : FiniteDimensional k (SimpleSum χ) := inferInstanceAs (FiniteDimensional k (ι → k)) |
| instance : Module (NewCorner k ι) (SimpleSum χ) := |
| Module.compHom (ι → k) (newAugmentation (k := k) (ι := ι)).toRingHom |
|
|
| def rightCharacter : Cᵐᵒᵖ →+* (ι → k) where |
| toFun c := χ c.unop |
| map_one' := map_one χ |
| map_zero' := map_zero χ |
| map_add' x y := map_add χ x.unop y.unop |
| map_mul' x y := by simp [map_mul, mul_comm] |
| |
| instance : Module Cᵐᵒᵖ (SimpleSum χ) := Module.compHom (ι → k) (rightCharacter χ) |
| |
| instance : SMulCommClass (NewCorner k ι) Cᵐᵒᵖ (SimpleSum χ) where |
| smul_comm e c x := by |
| change newAugmentation e * (rightCharacter χ c * value x) = |
| rightCharacter χ c * (newAugmentation e * value x) |
| exact mul_left_comm _ _ _ |
| |
| instance : IsScalarTower k (NewCorner k ι) (SimpleSum χ) where |
| smul_assoc r e x := by |
| change newAugmentation (r • e) * value x = r • (newAugmentation e * value x) |
| rw [map_smul, smul_mul_assoc] |
| |
| instance : IsScalarTower k Cᵐᵒᵖ (SimpleSum χ) where |
| smul_assoc r c x := by |
| change χ (r • c.unop) * value x = r • (χ c.unop * value x) |
| rw [map_smul, smul_mul_assoc] |
| end SimpleSum |
|
|
| variable (χ : C →ₐ[k] (ι → k)) |
| abbrev T := TriangularConstruction.Algebra C (NewCorner k ι) (SimpleSum χ) |
| |
| instance : Algebra k (T χ) := inferInstance |
| instance [Finite ι] [FiniteDimensional k C] : FiniteDimensional k (T χ) := |
| TriangularConstruction.finiteDimensional C (NewCorner k ι) (SimpleSum χ) k |
|
|
| def augmentation : T χ →ₐ[k] (Sum ι ι → k) where |
| toFun x := Sum.elim (χ x.fst.1) (newAugmentation x.fst.2) |
| map_one' := by ext i; cases i <;> simp [newAugmentation] |
| map_zero' := by ext i; cases i <;> simp [newAugmentation] |
| map_add' x y := by ext i; cases i <;> simp [newAugmentation, map_add] |
| map_mul' x y := by ext i; cases i <;> simp [newAugmentation, map_mul] |
| commutes' r := by ext i; cases i <;> simp [newAugmentation, Algebra.algebraMap_eq_smul_one] |
|
|
| lemma augmentation_surjective (hχ : Function.Surjective χ) : Function.Surjective (augmentation χ) := by |
| intro v |
| obtain ⟨c, hc⟩ := hχ (v ∘ Sum.inl) |
| refine ⟨((c, fun i => TrivSqZeroExt.inl (v (.inr i))), 0), ?_⟩ |
| ext i |
| cases i with |
| | inl i => exact congrFun hc i |
| | inr i => rfl |
|
|
| variable [DecidableEq ι] |
|
|
| def socle : Sum ι ι → T χ |
| | .inl i => (0, Pi.single i 1) |
| | .inr i => ((0, Pi.single i DualNumber.eps), 0) |
|
|
| lemma socle_ne_zero (i : Sum ι ι) : socle χ i ≠ 0 := by |
| intro h |
| cases i with |
| | inl i => |
| have := congrArg (fun x : T χ => (TriangularConstruction.offValue x.snd : ι → k) i) h |
| change (Pi.single i (1 : k) : ι → k) i = 0 at this |
| simp at this |
| | inr i => |
| have := congrArg (fun x : T χ => (x.fst.2 i).snd) h |
| simp [socle] at this |
|
|
| lemma socle_mul (i : Sum ι ι) (x : T χ) : |
| socle χ i * x = augmentation χ x i • socle χ i := by |
| cases i with |
| | inl i => |
| apply TrivSqZeroExt.ext |
| · change (0 : C × NewCorner k ι) * x.fst = _ • 0 |
| simp |
| · change newAugmentation (0 : NewCorner k ι) * (SimpleSum.value (TriangularConstruction.offValue x.snd)) + |
| SimpleSum.rightCharacter χ (MulOpposite.op x.fst.1) * (Pi.single i 1 : ι → k) = |
| χ x.fst.1 i • (Pi.single i 1 : ι → k) |
| ext j |
| by_cases h : j = i |
| · subst j; simp [SimpleSum.rightCharacter] |
| · simp [h] |
| | inr i => |
| apply TrivSqZeroExt.ext |
| · apply Prod.ext |
| · change (0 : C) * x.fst.1 = _ • 0 |
| simp |
| · change (Pi.single i DualNumber.eps : ι → DualNumber k) * x.fst.2 = |
| (x.fst.2 i).fst • (Pi.single i DualNumber.eps : ι → DualNumber k) |
| ext j : 1 |
| by_cases h : j = i |
| · subst j |
| simp only [Pi.mul_apply, Pi.smul_apply, Pi.single_eq_same] |
| apply TrivSqZeroExt.ext <;> simp |
| · simp [h] |
| · apply funext |
| intro j |
| change (newAugmentation (Pi.single i (DualNumber.eps : DualNumber k))) j * |
| (SimpleSum.value (TriangularConstruction.offValue x.snd)) j + χ x.fst.1 j * 0 = |
| (x.fst.2 i).fst * 0 |
| have he : newAugmentation (Pi.single i (DualNumber.eps : DualNumber k)) = (0 : ι → k) := by |
| ext j |
| by_cases h : j = i |
| · subst j; simp [newAugmentation] |
| · simp [newAugmentation, h] |
| simp [he] |
|
|
| end CummingsConstruction |
| end LittleFinitistic |
| end |
| end |
| end |
|
|
| end OAI |
|
|
