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7f5fab8 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 | 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
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