| import OAI.AlgebraicGeometry.StableCoordinate.FrameScaling |
| import OAI.AlgebraicGeometry.StableCoordinate.CoactionLND |
|
|
| namespace OAI |
|
|
| /-! |
| # Coassociativity of normalized determinant-frame lifts |
|
|
| A polynomial algebra map with the identity specialization has the additive |
| group law on all four frame entries as soon as it has that law on the three |
| quadric products. This is proved as an equality of two-parameter polynomial |
| frames, not by evaluating at field-valued parameters. |
| -/ |
|
|
| noncomputable section |
|
|
| namespace StableCoordinate |
| namespace FrameLift |
|
|
| open Polynomial |
|
|
| variable {R : Type*} [CommRing R] |
|
|
| / |
| def iterateHom (β : R →+* R[X]) : R →+* R[X][X] := |
| (Polynomial.mapRingHom β).comp β |
|
|
| / |
| def addParameterHom (β : R →+* R[X]) : R →+* R[X][X] := |
| (Polynomial.taylorAlgHom (X : R[X])).toRingHom.comp |
| ((Polynomial.mapRingHom (C : R →+* R[X])).comp β) |
|
|
| / |
| def CoassociativeAt (β : R →+* R[X]) (r : R) : Prop := |
| (β r).map β = taylor (X : R[X]) ((β r).map (C : R →+* R[X])) |
|
|
| @[simp] |
| theorem iterateHom_apply (β : R →+* R[X]) (r : R) : |
| iterateHom β r = (β r).map β := rfl |
|
|
| @[simp] |
| theorem addParameterHom_apply (β : R →+* R[X]) (r : R) : |
| addParameterHom β r = taylor (X : R[X]) ((β r).map (C : R →+* R[X])) := rfl |
|
|
| / |
| theorem outer_constantCoeff_agrees (β : R →+* R[X]) |
| (hβ : ∀ r, (β r).coeff 0 = r) (r : R) : |
| Polynomial.constantCoeff (iterateHom β r) = |
| Polynomial.constantCoeff (addParameterHom β r) := by |
| change ((β r).map β).coeff 0 = |
| (taylor (X : R[X]) ((β r).map (C : R →+* R[X]))).coeff 0 |
| rw [Polynomial.coeff_map, hβ, CoactionLND.coeff_universal_translate] |
| simp |
|
|
| / |
| frames by normalized polynomial-frame uniqueness. -/ |
| theorem coassociative_frame [IsReduced R] |
| (β : R →+* R[X]) (hβ : ∀ r, (β r).coeff 0 = r) |
| (F : Frame R) |
| (hbase : CoassociativeAt β (F.a*F.b) ∧ |
| CoassociativeAt β (F.d*F.c) ∧ CoassociativeAt β (F.d*F.b)) : |
| F.map (iterateHom β) = F.map (addParameterHom β) := by |
| let : IsReduced R[X] := ⟨by |
| intro p hp |
| ext n |
| simpa using isNilpotent_iff_eq_zero.mp ((Polynomial.isNilpotent_iff.mp hp) n)⟩ |
| apply normalized_univariate_unique |
| · change iterateHom β F.a * iterateHom β F.b = |
| addParameterHom β F.a * addParameterHom β F.b ∧ |
| iterateHom β F.d * iterateHom β F.c = |
| addParameterHom β F.d * addParameterHom β F.c ∧ |
| iterateHom β F.d * iterateHom β F.b = |
| addParameterHom β F.d * addParameterHom β F.b |
| simpa only [iterateHom_apply, addParameterHom_apply, CoassociativeAt, |
| map_mul, Polynomial.map_mul, Polynomial.taylor_mul] using hbase |
| · apply Frame.ext <;> exact outer_constantCoeff_agrees β hβ _ |
|
|
| / |
| theorem coassociative_on_frame_entries [IsReduced R] |
| (β : R →+* R[X]) (hβ : ∀ r, (β r).coeff 0 = r) |
| (F : Frame R) |
| (hbase : CoassociativeAt β (F.a*F.b) ∧ |
| CoassociativeAt β (F.d*F.c) ∧ CoassociativeAt β (F.d*F.b)) : |
| CoassociativeAt β F.a ∧ CoassociativeAt β F.d ∧ |
| CoassociativeAt β F.b ∧ CoassociativeAt β F.c := by |
| have h := coassociative_frame β hβ F hbase |
| exact ⟨congrArg Frame.a h, congrArg Frame.d h, congrArg Frame.b h, congrArg Frame.c h⟩ |
|
|
| end FrameLift |
| end StableCoordinate |
|
|
| end |
|
|
| end OAI |
|
|
