lean/OAI/AlgebraicGeometry/CommutingDerivations/ParameterRetraction.lean

import OAI.AlgebraicGeometry.AbhyankarSathaye.Identities
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.RingTheory.Localization.Away.Basic

namespace OAI

/-!
An explicit retraction proves saturation of the subring generated by `F + 1`.
-/
noncomputable section
namespace AbhyankarSathaye.CommutingDerivations

/-- A retract of a domain is saturated by powers of any nonzero element of
the retract. The maps and their left-inverse law are explicit hypotheses. -/
theorem retract_power_saturated {A B : Type*} [CommRing A] [IsDomain A]
    [CommRing B] (ι : B →+* A) (ρ : A →+* B)
    (hρι : ∀ b, ρ (ι b) = b) (b q : B) (a : A) (n : ℕ)
    (hb : ι b ≠ 0) (he : (ι b)^n * a = ι q) : a = ι (ρ a) := by
  apply mul_left_cancel₀ (pow_ne_zero n hb)
  calc
    (ι b)^n * a = ι q := he
    _ = (ι b)^n * ι (ρ a) := by
      have hh := congrArg ι (congrArg ρ he)
      simpa only [map_mul, map_pow, hρι] using hh.symm

/-- The parameter inverted in the localization, `F + 1`. -/
def ambientParameter : R := F + 1

/-- Restriction to the line `(h,u,v,w)=(t,0,0,0)`. -/
def parameterSlice : R →ₐ[ℂ] Polynomial ℂ :=
  MvPolynomial.aeval (fun i : Fin 4 => if i = 0 then Polynomial.X else 0)

@[simp] theorem parameterSlice_parameter : parameterSlice ambientParameter = Polynomial.X := by
  simp [parameterSlice, ambientParameter, F, p, P, x, y, s, S, h, u, v, w]

theorem parameterSlice_aeval :
    parameterSlice.comp (Polynomial.aeval ambientParameter) = AlgHom.id ℂ (Polynomial ℂ) := by
  apply Polynomial.algHom_ext
  simp

@[simp] theorem parameterSlice_aeval_apply (q : Polynomial ℂ) :
    parameterSlice (Polynomial.aeval ambientParameter q) = q := by
  exact AlgHom.congr_fun parameterSlice_aeval q

theorem parameter_aeval_injective :
    Function.Injective (Polynomial.aeval ambientParameter : Polynomial ℂ →ₐ[ℂ] R) :=
  (show Function.LeftInverse parameterSlice
    (Polynomial.aeval ambientParameter : Polynomial ℂ →ₐ[ℂ] R) from
      parameterSlice_aeval_apply).injective

theorem parameter_ne_zero : ambientParameter ≠ 0 := by
  intro hh
  have he := congrArg parameterSlice hh
  simp at he

/-- Exact polynomial witness for descent from a Laurent expression. -/
theorem parameter_power_saturated (a : R) (q : Polynomial ℂ) (n : ℕ)
    (he : ambientParameter^n * a = Polynomial.aeval ambientParameter q) :
    a = Polynomial.aeval ambientParameter (parameterSlice a) := by
  apply mul_left_cancel₀ (pow_ne_zero n parameter_ne_zero)
  have hh := congrArg (Polynomial.aeval ambientParameter : Polynomial ℂ →ₐ[ℂ] R)
    (congrArg parameterSlice he)
  have hh' : ambientParameter^n * Polynomial.aeval ambientParameter (parameterSlice a) =
      Polynomial.aeval ambientParameter q := by
    simpa only [map_mul, map_pow, parameterSlice_parameter,
      parameterSlice_aeval_apply, Polynomial.aeval_X] using hh
  exact he.trans hh'.symm

theorem parameter_localization_injective (L : Type*) [CommRing L] [Algebra R L]
    [IsLocalization.Away ambientParameter L] : Function.Injective (algebraMap R L) := by
  apply IsLocalization.injective L (M := Submonoid.powers ambientParameter)
  intro z hz
  obtain ⟨n, rfl⟩ := hz
  exact mem_nonZeroDivisors_of_ne_zero (pow_ne_zero n parameter_ne_zero)

/-- Exact intersection with the ambient ring, in denominator-cleared form.
The left side says that the localized image has a Laurent expression in `c`.
The right side supplies an ordinary polynomial expression in `c`.
There is no fraction-field-intersection hypothesis. -/
theorem parameter_laurent_intersection (L : Type*) [CommRing L] [Algebra R L]
    [IsLocalization.Away ambientParameter L] (a : R) :
    (∃ (n : ℕ) (q : Polynomial ℂ),
      (algebraMap R L ambientParameter)^n * algebraMap R L a =
        algebraMap R L (Polynomial.aeval ambientParameter q)) ↔
    ∃ q : Polynomial ℂ, a = Polynomial.aeval ambientParameter q := by
  constructor
  · rintro ⟨n, q, hq⟩
    refine ⟨parameterSlice a, parameter_power_saturated a q n ?_⟩
    apply parameter_localization_injective L
    simpa only [map_mul, map_pow] using hq
  · rintro ⟨q, rfl⟩
    exact ⟨0, q, by simp⟩

end AbhyankarSathaye.CommutingDerivations

end

end OAI