| import OAI.Algebra.Finitistic.Localization.NormalizedQuotientAction |
| import Mathlib.Algebra.FreeAlgebra |
| import Mathlib.Algebra.RingQuot |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory |
| namespace LittleFinitistic.QuadraticEncoding |
| universe u v |
| variable {k : Type u} [Field k] {d t : ℕ} (P : Relations k d t) |
|
|
| / |
| def relationPolynomial (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 |
|
|
| def presentationRelation (a b : FreeAlgebra k (Fin d)) : Prop := |
| ∃ j, a = relationPolynomial P j ∧ b = 0 |
|
|
| / |
| abbrev PresentedAlgebra := RingQuot (presentationRelation P) |
| def presentedGenerator (h : Fin d) : PresentedAlgebra P := |
| RingQuot.mkAlgHom k (presentationRelation P) (FreeAlgebra.ι k h) |
|
|
| theorem presentedGenerator_satisfies : P.Satisfies (presentedGenerator P) := by |
| intro j |
| have h := RingQuot.mkAlgHom_rel k (s := presentationRelation P) ⟨j,rfl,rfl⟩ |
| simpa only [relationPolynomial,map_add,map_sum,map_smul,map_mul,map_one,map_zero, |
| presentedGenerator] using h |
|
|
| variable {A : Type v} [Ring A] [Algebra k A] |
|
|
| / |
| def presentationLift (x : Fin d → A) (hx : P.Satisfies x) : PresentedAlgebra P →ₐ[k] A := |
| RingQuot.liftAlgHom k ⟨FreeAlgebra.lift k x, by |
| rintro a b ⟨j,rfl,rfl⟩ |
| simpa only [relationPolynomial,map_add,map_sum,map_smul,map_mul,map_one,map_zero, |
| FreeAlgebra.lift_ι_apply] using hx j⟩ |
|
|
| @[simp] theorem presentationLift_generator (x : Fin d → A) (hx : P.Satisfies x) (h : Fin d) : |
| presentationLift P x hx (presentedGenerator P h) = x h := by |
| simp [presentationLift,presentedGenerator,RingQuot.liftAlgHom_mkAlgHom_apply] |
|
|
| / |
| def quotientAction : PresentedAlgebra P →ₐ[k] End (vertexQ P 0) := |
| presentationLift P (normalizedGenerator P) (normalizedGenerator_satisfies P) |
|
|
| @[simp] theorem quotientAction_generator (h : Fin d) : |
| quotientAction P (presentedGenerator P h) = normalizedGenerator P h := |
| presentationLift_generator _ _ _ _ |
|
|
| / |
| def quotientActionOfPresentation (e : PresentedAlgebra P ≃ₐ[k] A) : |
| A →ₐ[k] End (vertexQ P 0) := (quotientAction P).comp e.symm.toAlgHom |
|
|
| @[simp] theorem quotientActionOfPresentation_generator (e : PresentedAlgebra P ≃ₐ[k] A) |
| (h : Fin d) : quotientActionOfPresentation P e (e (presentedGenerator P h)) = |
| normalizedGenerator P h := by simp [quotientActionOfPresentation] |
| end LittleFinitistic.QuadraticEncoding |
|
|
| end |
|
|
| end OAI |
|
|
