| import Mathlib.CategoryTheory.Linear.Basic |
| import OAI.Algebra.Finitistic.Encoding.QuadraticEncoding |
| import OAI.Algebra.Finitistic.Encoding.QuadraticPresentation |
|
|
| namespace OAI |
|
|
| namespace LittleFinitistic.QuotientAction |
| open CategoryTheory CategoryTheory.Category CategoryTheory.Limits CategoryTheory.Preadditive |
| open QuadraticEncoding |
|
|
| variable {k : Type*} [Field k] {C : Type*} [Category C] [Preadditive C] [Linear k C] |
| variable {X₀ X₁ X₂ : C} {d t : ℕ} (P : Relations k d t) |
| (s : X₀ ⟶ X₁) (s' : X₁ ⟶ X₂) [IsIso s] [IsIso s'] |
| (a : Fin d → (X₀ ⟶ X₁)) (b : Fin d → (X₁ ⟶ X₂)) |
| (hc : ∀ h, s ≫ b h = a h ≫ s') |
|
|
| noncomputable def normalized (h : Fin d) : End X₀ := a h ≫ inv s |
|
|
| include hc in |
| omit [Preadditive C] [IsIso s'] in |
| lemma normalized_product_comp (h l : Fin d) : |
| (normalized s a h * normalized s a l) ≫ s ≫ s' = a l ≫ b h := by |
| simp only [normalized,End.mul_def,assoc,IsIso.inv_hom_id_assoc] |
| rw [← hc h] |
| simp only [IsIso.inv_hom_id_assoc] |
|
|
| omit [Preadditive C] [IsIso s'] in |
| lemma normalized_comp (h : Fin d) : normalized s a h ≫ s ≫ s' = a h ≫ s' := by |
| simp [normalized] |
|
|
| variable (hr : ∀ j, |
| (∑ h, ∑ l, P.quadratic j h l • (a l ≫ b h)) + |
| (∑ h, P.linear j h • (a h ≫ s')) + P.constant j • (s ≫ s') = 0) |
|
|
| include hc hr in |
| / |
| theorem normalized_relations (j : Fin t) : |
| (∑ h, ∑ l, P.quadratic j h l • (normalized s a h * normalized s a l)) + |
| (∑ h, P.linear j h • normalized s a h) + P.constant j • (1 : End X₀) = 0 := by |
| apply (cancel_mono (s ≫ s')).mp |
| dsimp only [End] |
| simp only [Preadditive.add_comp,Preadditive.sum_comp,Linear.smul_comp,End.one_def,id_comp,zero_comp] |
| simpa only [← End.mul_def,normalized_product_comp s s' a b hc,normalized_comp] using hr j |
|
|
| / |
| def polynomial (j : Fin t) : FreeAlgebra k (Fin d) := |
| (∑ h, ∑ l, P.quadratic j h l • (FreeAlgebra.ι k h * FreeAlgebra.ι k l)) + |
| (∑ h, P.linear j h • FreeAlgebra.ι k h) + P.constant j • 1 |
|
|
| noncomputable def freeAction : FreeAlgebra k (Fin d) →ₐ[k] End X₀ := |
| FreeAlgebra.lift k (normalized s a) |
|
|
| include hc hr in |
| lemma freeAction_polynomial (j : Fin t) : freeAction s a (polynomial P j) = 0 := by |
| simpa only [freeAction,polynomial,map_add,map_sum,map_smul,map_mul,map_one, |
| FreeAlgebra.lift_ι_apply] using normalized_relations P s s' a b hc hr j |
|
|
| / |
| noncomputable def quotientAction : |
| QuadraticPresentation.Original k (Fin d) (polynomial P) →ₐ[k] End X₀ := |
| (QuadraticPresentation.original k (Fin d) (polynomial P)).liftₐ (freeAction s a) |
| (RingCon.ringConGen_le.mpr (by |
| rintro x y ⟨j,rfl,rfl⟩ |
| change freeAction s a (polynomial P j) = freeAction s a 0 |
| rw [map_zero,freeAction_polynomial P s s' a b hc hr])) |
|
|
| @[simp] theorem quotientAction_generator (h : Fin d) : |
| quotientAction P s s' a b hc hr |
| (QuadraticPresentation.originalMap k (Fin d) (polynomial P) (FreeAlgebra.ι k h)) = |
| normalized s a h := by |
| simp [quotientAction,QuadraticPresentation.originalMap,freeAction] |
|
|
| / |
| noncomputable def presentedAction {R : Type*} [Ring R] [Algebra k R] |
| (E : QuadraticPresentation.Original k (Fin d) (polynomial P) ≃ₐ[k] R) : |
| R →ₐ[k] End X₀ := (quotientAction P s s' a b hc hr).comp E.symm.toAlgHom |
|
|
| omit [Preadditive C] in |
| / |
| theorem normalized_evaluation {D : Type*} [Category D] (F : C ⥤ D) |
| {Y : D} (e₀ : F.obj X₀ ≅ Y) (e₁ : F.obj X₁ ≅ Y) |
| (x : Fin d → (Y ⟶ Y)) |
| (hs : F.map s ≫ e₁.hom = e₀.hom) |
| (ha : ∀ h, F.map (a h) ≫ e₁.hom = e₀.hom ≫ x h) (h : Fin d) : |
| F.map (normalized s a h) ≫ e₀.hom = e₀.hom ≫ x h := by |
| have hsi : F.map (inv s) ≫ e₀.hom = e₁.hom := by |
| apply (cancel_epi (F.map s)).mp |
| rw [← assoc,← F.map_comp,IsIso.hom_inv_id,F.map_id,id_comp,hs] |
| change F.map (a h ≫ inv s) ≫ e₀.hom = _ |
| rw [F.map_comp,assoc,hsi,ha] |
| end LittleFinitistic.QuotientAction |
|
|
| end OAI |
|
|
