| 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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|
