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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 | import OAI.AlgebraicGeometry.CommutingDerivations.Main
import OAI.AlgebraicGeometry.CommutingDerivations.ExtendedExtraPartials
import OAI.AlgebraicGeometry.CommutingDerivations.MixedDescentFamily
namespace OAI
/-! The all-dimensional family may be chosen with the additional n-4
directions literally equal to the ordinary ambient coordinate partials. -/
noncomputable section
namespace AbhyankarSathaye.CommutingDerivations
open MvPolynomial
def withOrdinaryExtras {n : ℕ} (hn : 4 ≤ n)
(d : Fin (n-1) → Derivation ℂ (MvPolynomial (Fin n) ℂ) (MvPolynomial (Fin n) ℂ)) :
Fin (n-1) → Derivation ℂ (MvPolynomial (Fin n) ℂ) (MvPolynomial (Fin n) ℂ) :=
fun k => match (blocksToOutput hn).symm k with
| Sum.inl i => pderiv (blocksToAmbient hn (Sum.inl i))
| Sum.inr _ => d k
def extraClearingExponent {n : ℕ} (hn : 4 ≤ n) (N : ℕ) (k : Fin (n-1)) : ℕ :=
match (blocksToOutput hn).symm k with
| Sum.inl _ => 0
| Sum.inr _ => N
theorem commuting_derivations_with_ordinary_extras (n : ℕ) (hn : 4 ≤ n) :
∃ d : Fin (n-1) → Derivation ℂ (MvPolynomial (Fin n) ℂ) (MvPolynomial (Fin n) ℂ),
(∀ i, d i (extendedF hn + 1) = 0) ∧
(∀ i j p, d i (d j p) = d j (d i p)) ∧
(∀ i, LocallyNilpotent (d i)) ∧
LinearIndependent (MvPolynomial (Fin n) ℂ) d ∧
(∀ p, (∀ i, d i p = 0) ↔
p ∈ Algebra.adjoin ℂ ({extendedF hn} : Set (MvPolynomial (Fin n) ℂ))) ∧
(∀ g, (∀ i, d i g = 0) →
¬ ∃ (e : MvPolynomial (Fin n) ℂ ≃ₐ[ℂ] MvPolynomial (Fin n) ℂ) (j : Fin n),
e (X j) = g) ∧
(∀ i : Fin (n-4), d (blocksToOutput hn (Sum.inl i)) =
pderiv (blocksToAmbient hn (Sum.inl i))) := by
let := extendedAlgebra hn
let := extendedScalarTower hn
let := extended_isLocalization hn
let : IsLocalization.Away (extendedAmbientParameter hn) (ExtendedLocalizedAmbient n) := by
change IsLocalization.Away (extendedF hn+1) (ExtendedLocalizedAmbient n)
exact extended_isLocalization hn
let ι : MvPolynomial (Fin n) ℂ →ₐ[ℂ] ExtendedLocalizedAmbient n := extendedLocAlgHom hn
have hinj : Function.Injective ι :=
extendedParameter_localization_injective hn (ExtendedLocalizedAmbient n)
have hu : IsUnit (ι (extendedF hn+1)) :=
IsLocalization.Away.algebraMap_isUnit (extendedF hn+1)
obtain ⟨N, d, hdesc, _, _, _, _, _⟩ :=
exists_commuting_family_of_localization (extendedF hn + 1) hinj
(extendedLocalizationEquiv hn) coefficientC (extendedLocalizationEquiv_c hn)
(Algebra.adjoin ℂ ({extendedF hn} : Set (MvPolynomial (Fin n) ℂ)))
(extended_localization_kernel_intersection hn)
have hdesc' (k : Fin (n-1)) (p : MvPolynomial (Fin n) ℂ) :
ι (withOrdinaryExtras hn d k p) =
(ι (extendedF hn+1))^(extraClearingExponent hn N k) *
localizedPartial (extendedLocalizationEquiv hn) k (ι p) := by
obtain ⟨j, rfl⟩ := (blocksToOutput hn).surjective k
cases j with
| inl i =>
simp only [withOrdinaryExtras, extraClearingExponent,
Equiv.symm_apply_apply, pow_zero, one_mul]
exact (localized_extra_is_ordinary_partial hn i p).symm
| inr j =>
simp only [withOrdinaryExtras, extraClearingExponent,
Equiv.symm_apply_apply]
exact hdesc (blocksToOutput hn (Sum.inr j)) p
have hfrac (j : Fin (n-1)) : ∃ (m : ℕ) (r : MvPolynomial (Fin n) ℂ),
(extendedLocalizationEquiv hn).symm (X j) * (ι (extendedF hn+1))^m = ι r :=
IsLocalization.Away.surj (extendedF hn+1) ((extendedLocalizationEquiv hn).symm (X j))
obtain ⟨hfix, hcomm, hnil, hind, hker⟩ :=
family_properties_of_mixed_descent ι hinj (extendedF hn+1) hu
(extendedLocalizationEquiv hn) coefficientC (extendedLocalizationEquiv_c hn)
(extraClearingExponent hn N) (withOrdinaryExtras hn d) hdesc' hfrac
(Algebra.adjoin ℂ ({extendedF hn} : Set (MvPolynomial (Fin n) ℂ)))
(extended_localization_kernel_intersection hn)
refine ⟨withOrdinaryExtras hn d, hfix, hcomm, hnil, hind, hker, ?_, ?_⟩
· intro g hg
apply element_of_extendedF_subring_not_coordinate hn g
have hmem := (hker g).mp hg
rw [Algebra.adjoin_singleton_eq_range_aeval] at hmem
obtain ⟨q, hq⟩ := hmem
exact ⟨q, hq.symm⟩
· intro i
simp [withOrdinaryExtras]
end AbhyankarSathaye.CommutingDerivations
end
end OAI
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