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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 | 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
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