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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 | import OAI.AlgebraicGeometry.CommutingDerivations.ParameterRetraction
import OAI.AlgebraicGeometry.AbhyankarSathaye.Stabilization
import Mathlib.RingTheory.Adjoin.Polynomial.Basic
namespace OAI
/-! The explicit intersection proof retains every dimension n >= 4 and
identifies its ordinary polynomial side with the standard subalgebra C[F_n]. -/
noncomputable section
namespace AbhyankarSathaye.CommutingDerivations
def extendedAmbientParameter {n : ℕ} (hn : 4 ≤ n) : MvPolynomial (Fin n) ℂ :=
extendedF hn + 1
def extendedParameterSlice {n : ℕ} (_hn : 4 ≤ n) :
MvPolynomial (Fin n) ℂ →ₐ[ℂ] Polynomial ℂ :=
MvPolynomial.aeval (fun i : Fin n => if i.val = 0 then Polynomial.X else 0)
theorem extendedParameterSlice_rename {n : ℕ} (hn : 4 ≤ n) (q : R) :
extendedParameterSlice hn (MvPolynomial.rename (fourInclusion hn) q) = parameterSlice q := by
unfold extendedParameterSlice parameterSlice
rw [MvPolynomial.aeval_rename]
have he : (fun i : Fin n => if i.val = 0 then (Polynomial.X : Polynomial ℂ) else 0) ∘
fourInclusion hn = (fun i : Fin 4 => if i = 0 then Polynomial.X else 0) := by
funext i
change (if i.val = 0 then (Polynomial.X : Polynomial ℂ) else 0) =
(if i = 0 then Polynomial.X else 0)
by_cases hi : i = 0
· subst i
rfl
· have hv : i.val ≠ 0 := fun hh => hi (Fin.ext hh)
rw [ite_eq_right hv, ite_eq_right hi]
rw [he]
@[simp] theorem extendedParameterSlice_parameter {n : ℕ} (hn : 4 ≤ n) :
extendedParameterSlice hn (extendedAmbientParameter hn) = Polynomial.X := by
have he : extendedAmbientParameter hn = MvPolynomial.rename (fourInclusion hn) ambientParameter := by
simp [extendedAmbientParameter, extendedF, ambientParameter]
rw [he, extendedParameterSlice_rename, parameterSlice_parameter]
theorem extendedParameterSlice_aeval {n : ℕ} (hn : 4 ≤ n) :
(extendedParameterSlice hn).comp (Polynomial.aeval (extendedAmbientParameter hn)) =
AlgHom.id ℂ (Polynomial ℂ) := by
apply Polynomial.algHom_ext
simp
@[simp] theorem extendedParameterSlice_aeval_apply {n : ℕ} (hn : 4 ≤ n) (q : Polynomial ℂ) :
extendedParameterSlice hn (Polynomial.aeval (extendedAmbientParameter hn) q) = q :=
AlgHom.congr_fun (extendedParameterSlice_aeval hn) q
theorem extendedParameter_aeval_injective {n : ℕ} (hn : 4 ≤ n) :
Function.Injective (Polynomial.aeval (extendedAmbientParameter hn) :
Polynomial ℂ →ₐ[ℂ] MvPolynomial (Fin n) ℂ) :=
(show Function.LeftInverse (extendedParameterSlice hn)
(Polynomial.aeval (extendedAmbientParameter hn) :
Polynomial ℂ →ₐ[ℂ] MvPolynomial (Fin n) ℂ) from
extendedParameterSlice_aeval_apply hn).injective
theorem extendedParameter_ne_zero {n : ℕ} (hn : 4 ≤ n) : extendedAmbientParameter hn ≠ 0 := by
intro hh
have he := congrArg (extendedParameterSlice hn) hh
simp at he
theorem extendedParameter_power_saturated {n : ℕ} (hn : 4 ≤ n)
(a : MvPolynomial (Fin n) ℂ) (q : Polynomial ℂ) (m : ℕ)
(he : (extendedAmbientParameter hn)^m * a = Polynomial.aeval (extendedAmbientParameter hn) q) :
a = Polynomial.aeval (extendedAmbientParameter hn) (extendedParameterSlice hn a) := by
apply mul_left_cancel₀ (pow_ne_zero m (extendedParameter_ne_zero hn))
have hh := congrArg (Polynomial.aeval (extendedAmbientParameter hn) :
Polynomial ℂ →ₐ[ℂ] MvPolynomial (Fin n) ℂ)
(congrArg (extendedParameterSlice hn) he)
have hh' : (extendedAmbientParameter hn)^m *
Polynomial.aeval (extendedAmbientParameter hn) (extendedParameterSlice hn a) =
Polynomial.aeval (extendedAmbientParameter hn) q := by
simpa only [map_mul, map_pow, extendedParameterSlice_parameter,
extendedParameterSlice_aeval_apply, Polynomial.aeval_X] using hh
exact he.trans hh'.symm
theorem extendedParameter_localization_injective {n : ℕ} (hn : 4 ≤ n)
(L : Type*) [CommRing L] [Algebra (MvPolynomial (Fin n) ℂ) L]
[IsLocalization.Away (extendedAmbientParameter hn) L] :
Function.Injective (algebraMap (MvPolynomial (Fin n) ℂ) L) := by
apply IsLocalization.injective L (M := Submonoid.powers (extendedAmbientParameter hn))
intro z hz
obtain ⟨m, rfl⟩ := hz
exact mem_nonZeroDivisors_of_ne_zero (pow_ne_zero m (extendedParameter_ne_zero hn))
/-- Replacing f by f+1 does not change the literal polynomial-expression set. -/
theorem exists_aeval_add_one_iff {K A : Type*} [CommRing K] [CommRing A] [Algebra K A]
(f a : A) :
(∃ q : Polynomial K, a = Polynomial.aeval (f + 1) q) ↔
∃ q : Polynomial K, a = Polynomial.aeval f q := by
constructor
· rintro ⟨q, rfl⟩
refine ⟨q.comp (Polynomial.X + 1), ?_⟩
simp [Polynomial.aeval_comp]
· rintro ⟨q, rfl⟩
refine ⟨q.comp (Polynomial.X - 1), ?_⟩
simp [Polynomial.aeval_comp]
/-- Complete dimension-quantified intersection in the standard subalgebra
representation of C[F_n], with localization membership given by one Laurent
denominator and an ordinary polynomial numerator. -/
theorem extendedParameter_laurent_intersection {n : ℕ} (hn : 4 ≤ n)
(L : Type*) [CommRing L] [Algebra (MvPolynomial (Fin n) ℂ) L]
[IsLocalization.Away (extendedAmbientParameter hn) L] (a : MvPolynomial (Fin n) ℂ) :
(∃ (m : ℕ) (q : Polynomial ℂ),
(algebraMap (MvPolynomial (Fin n) ℂ) L (extendedAmbientParameter hn))^m *
algebraMap (MvPolynomial (Fin n) ℂ) L a =
algebraMap (MvPolynomial (Fin n) ℂ) L (Polynomial.aeval (extendedAmbientParameter hn) q)) ↔
a ∈ Algebra.adjoin ℂ ({extendedF hn} : Set (MvPolynomial (Fin n) ℂ)) := by
have hex :
(∃ (m : ℕ) (q : Polynomial ℂ),
(algebraMap (MvPolynomial (Fin n) ℂ) L (extendedAmbientParameter hn))^m *
algebraMap (MvPolynomial (Fin n) ℂ) L a =
algebraMap (MvPolynomial (Fin n) ℂ) L (Polynomial.aeval (extendedAmbientParameter hn) q)) ↔
∃ q : Polynomial ℂ, a = Polynomial.aeval (extendedAmbientParameter hn) q := by
constructor
· rintro ⟨m, q, hq⟩
refine ⟨extendedParameterSlice hn a, extendedParameter_power_saturated hn a q m ?_⟩
apply extendedParameter_localization_injective hn L
simpa only [map_mul, map_pow] using hq
· rintro ⟨q, rfl⟩
exact ⟨0, q, by simp⟩
rw [hex, extendedAmbientParameter, exists_aeval_add_one_iff,
Algebra.adjoin_singleton_eq_range_aeval]
change (∃ q : Polynomial ℂ, a = Polynomial.aeval (extendedF hn) q) ↔
∃ q : Polynomial ℂ, Polynomial.aeval (extendedF hn) q = a
simp only [eq_comm]
end AbhyankarSathaye.CommutingDerivations
end
end OAI
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