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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 | import OAI.AlgebraicGeometry.CommutingDerivations.CoefficientInstances
import OAI.AlgebraicGeometry.CommutingDerivations.ExtendedLocalization
import OAI.AlgebraicGeometry.CommutingDerivations.ExtendedKernelIntersection
import OAI.AlgebraicGeometry.CommutingDerivations.FamilyFromLocalization
import OAI.AlgebraicGeometry.CommutingDerivations.KernelObstruction
namespace OAI
/-!
Commuting locally nilpotent derivations for the explicit Abhyankar polynomial
in every dimension `n ≥ 4`, with common kernel `ℂ[Fₙ]`.
-/
noncomputable section
namespace AbhyankarSathaye.CommutingDerivations
open MvPolynomial
theorem commuting_derivations_all_dimensions (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) := 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
have hinj : Function.Injective
(algebraMap (MvPolynomial (Fin n) ℂ) (ExtendedLocalizedAmbient n)) :=
extendedParameter_localization_injective hn (ExtendedLocalizedAmbient n)
obtain ⟨N, d, hdesc, hfix, hcomm, hnil, hind, hker⟩ :=
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)
refine ⟨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⟩
theorem extendedF_four : extendedF (by decide : 4 ≤ 4) = F := by
have hi : fourInclusion (by decide : 4 ≤ 4) = id := by
funext i
apply Fin.ext
rfl
unfold extendedF
rw [hi, rename_id]
rfl
/-- Three commuting, R-linearly independent locally nilpotent derivations of
the original four-variable ring, with exact common kernel C[F]. Every element
of this kernel, in particular any generator of it, fails to be a coordinate. -/
theorem commuting_derivations_dimension_four :
∃ d : Fin 3 → Derivation ℂ R R,
(∀ i, d i (F + 1) = 0) ∧
(∀ i j p, d i (d j p) = d j (d i p)) ∧
(∀ i, LocallyNilpotent (d i)) ∧
LinearIndependent R d ∧
(∀ p, (∀ i, d i p = 0) ↔ p ∈ Algebra.adjoin ℂ ({F} : Set R)) ∧
(∀ g, (∀ i, d i g = 0) → ¬ ∃ (e : R ≃ₐ[ℂ] R) (j : Fin 4), e (X j) = g) := by
simpa only [extendedF_four] using commuting_derivations_all_dimensions 4 (by decide)
end AbhyankarSathaye.CommutingDerivations
end
end OAI
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