| import OAI.AlgebraicGeometry.CommutingDerivations.ParameterBezout |
| import OAI.AlgebraicGeometry.AbhyankarSathaye.Reconstruction |
| import OAI.AlgebraicGeometry.AbhyankarSathaye.Quotient |
| import Mathlib.RingTheory.Localization.Away.Basic |
| import Mathlib.Algebra.Polynomial.AlgebraMap |
| import Mathlib.Data.Fin.VecNotation |
|
|
| namespace OAI |
|
|
| /-! |
| The Laurent coefficient algebra and localization coordinates. |
| The coefficient map is the lift of `t ↦ F + 1`. |
| -/ |
| noncomputable section |
| namespace AbhyankarSathaye.CommutingDerivations |
| open MvPolynomial |
|
|
| def ambientC : R := F + 1 |
| abbrev LocalizedAmbient := Localization.Away ambientC |
| abbrev LaurentCoefficients := Localization.Away (Polynomial.X : Polynomial ℂ) |
| abbrev RelativePlane := MvPolynomial (Fin 3) LaurentCoefficients |
|
|
| def loc : R →+* LocalizedAmbient := algebraMap R LocalizedAmbient |
| def coefficientC : LaurentCoefficients := |
| algebraMap (Polynomial ℂ) LaurentCoefficients Polynomial.X |
| def coefficientD : LaurentCoefficients := |
| IsLocalization.Away.invSelf (Polynomial.X : Polynomial ℂ) |
| def localD : LocalizedAmbient := IsLocalization.Away.invSelf ambientC |
|
|
| theorem coefficient_unit : coefficientC * coefficientD = 1 := |
| IsLocalization.Away.mul_invSelf _ |
|
|
| theorem local_unit : loc ambientC * localD = 1 := |
| IsLocalization.Away.mul_invSelf _ |
|
|
| def polynomialCoefficientMap : Polynomial ℂ →+* LocalizedAmbient := |
| (Polynomial.aeval (loc ambientC)).toRingHom |
|
|
| def coefficientMap : LaurentCoefficients →+* LocalizedAmbient := |
| IsLocalization.Away.lift Polynomial.X |
| (by |
| refine ⟨⟨polynomialCoefficientMap Polynomial.X, localD, ?_, ?_⟩, rfl⟩ |
| · simpa [polynomialCoefficientMap] using local_unit |
| · simpa [polynomialCoefficientMap, mul_comm] using local_unit) |
|
|
| @[simp] theorem coefficientMap_C : coefficientMap coefficientC = loc ambientC := by |
| simp [coefficientMap, coefficientC, polynomialCoefficientMap] |
|
|
| @[simp] theorem coefficientMap_D : coefficientMap coefficientD = localD := by |
| have h1 : loc ambientC * coefficientMap coefficientD = 1 := by |
| simpa using congrArg coefficientMap coefficient_unit |
| calc |
| coefficientMap coefficientD = (loc ambientC * localD) * coefficientMap coefficientD := by |
| rw [local_unit, one_mul] |
| _ = localD * (loc ambientC * coefficientMap coefficientD) := by ring |
| _ = localD := by rw [h1, mul_one] |
|
|
| def localAlpha : LocalizedAmbient := localD^2 * parameterA (loc ambientC) (loc x) (loc s) |
| def localBeta : LocalizedAmbient := |
| localD^2 * parameterB (loc ambientC) (loc x) (loc y) (loc s) |
| def localT : LocalizedAmbient := parameter localAlpha localBeta (loc u) (loc v) (loc w) |
| def localCoordinates : Fin 3 → LocalizedAmbient := ![loc shiftX, loc shiftY, localT] |
|
|
| theorem local_cusp : (loc x)^2 + (loc y)^3 = loc s * loc h := by |
| simpa [mul_comm] using congrArg loc cusp |
|
|
| theorem local_section : loc h = loc ambientC + P (loc x) (loc y) (loc s) := by |
| have he : h = ambientC + P x y s := by unfold ambientC F p; ring |
| simpa using congrArg loc he |
|
|
| theorem local_bezout : localAlpha * loc h + localBeta * loc y = 1 := |
| parameter_bezout_unit _ _ _ _ _ _ local_cusp local_section local_unit |
|
|
| theorem local_recover_s : localD * ((loc shiftX)^2 + (loc shiftY)^3) = loc s := by |
| simpa [shiftX, shiftY] using |
| parameter_recover_s _ _ _ _ _ _ local_cusp local_section local_unit |
|
|
| /-- Evaluation at the displayed X,Y,T, with the explicit relative coefficient map. -/ |
| def relativeBackward : RelativePlane →+* LocalizedAmbient := |
| eval₂Hom coefficientMap localCoordinates |
|
|
| @[simp] theorem relativeBackward_C (a : LaurentCoefficients) : |
| relativeBackward (C a) = coefficientMap a := by |
| simp [relativeBackward] |
|
|
| @[simp] theorem relativeBackward_X (i : Fin 3) : |
| relativeBackward (X i) = localCoordinates i := by |
| simp [relativeBackward] |
|
|
| /-- A relative algebra equivalence over `ℂ[c,c⁻¹]`, with `c = F + 1`, |
| and its coordinate equations. -/ |
| def RelativeLocalizationContract : Prop := |
| ∃ e : LocalizedAmbient ≃+* RelativePlane, |
| (∀ a, e (coefficientMap a) = C a) ∧ |
| e (loc shiftX) = X 0 ∧ e (loc shiftY) = X 1 ∧ e localT = X 2 ∧ |
| e.symm.toRingHom = relativeBackward |
|
|
| end AbhyankarSathaye.CommutingDerivations |
|
|
| end |
|
|
| end OAI |
|
|
