File size: 3,206 Bytes
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 | import OAI.AlgebraicGeometry.CommutingDerivations.ExtendedLocalization
import OAI.AlgebraicGeometry.CommutingDerivations.LocalizedPartials
namespace OAI
/-!
Ordinary partial derivatives in the unused variables agree with the corresponding
localized coordinate derivatives and descend without a clearing factor.
-/
noncomputable section
namespace AbhyankarSathaye.CommutingDerivations
open MvPolynomial
theorem pderiv_sumAlgEquiv_symm {K I J : Type*} [CommRing K]
(i : I) (p : MvPolynomial I (MvPolynomial J K)) :
pderiv (Sum.inl i) ((sumAlgEquiv K I J).symm p) =
(sumAlgEquiv K I J).symm (pderiv i p) := by
apply (sumAlgEquiv K I J).injective
simpa using (pderiv_sumAlgEquiv i ((sumAlgEquiv K I J).symm p)).symm
theorem splitAmbient_extra_pderiv {n : ℕ} (hn : 4 ≤ n)
(i : Fin (n-4)) (p : MvPolynomial (Fin n) ℂ) :
pderiv i (splitAmbient hn p) =
splitAmbient hn (pderiv (blocksToAmbient hn (Sum.inl i)) p) := by
have he := pderiv_rename (blocksToAmbient hn).symm.injective
(blocksToAmbient hn (Sum.inl i)) p
simp only [Equiv.symm_apply_apply] at he
change pderiv i
(sumAlgEquiv ℂ (Fin (n-4)) (Fin 4) (rename (blocksToAmbient hn).symm p)) =
sumAlgEquiv ℂ (Fin (n-4)) (Fin 4)
(rename (blocksToAmbient hn).symm (pderiv (blocksToAmbient hn (Sum.inl i)) p))
rw [pderiv_sumAlgEquiv, he]
theorem extendedLocMap_extra_pderiv {n : ℕ} (hn : 4 ≤ n)
(i : Fin (n-4)) (p : MvPolynomial (Fin n) ℂ) :
pderiv i (extendedLocMap hn p) =
extendedLocMap hn (pderiv (blocksToAmbient hn (Sum.inl i)) p) := by
change pderiv i (MvPolynomial.map loc (splitAmbient hn p)) =
MvPolynomial.map loc
(splitAmbient hn (pderiv (blocksToAmbient hn (Sum.inl i)) p))
rw [pderiv_map, splitAmbient_extra_pderiv]
theorem extendedLocalizationEquiv_extra_pderiv {n : ℕ} (hn : 4 ≤ n)
(i : Fin (n-4)) (p : ExtendedLocalizedAmbient n) :
pderiv (blocksToOutput hn (Sum.inl i)) (extendedLocalizationEquiv hn p) =
extendedLocalizationEquiv hn (pderiv i p) := by
change pderiv (blocksToOutput hn (Sum.inl i))
(rename (blocksToOutput hn)
((sumAlgEquiv LaurentCoefficients (Fin (n-4)) (Fin 3)).symm
(MvPolynomial.map localizationComplexEquiv p))) =
rename (blocksToOutput hn)
((sumAlgEquiv LaurentCoefficients (Fin (n-4)) (Fin 3)).symm
(MvPolynomial.map localizationComplexEquiv (pderiv i p)))
rw [pderiv_rename (blocksToOutput hn).injective, pderiv_sumAlgEquiv_symm, pderiv_map]
/-- The transported derivative indexed by an added coordinate agrees exactly
with the ordinary partial derivative in the full original n-variable ring. -/
theorem localized_extra_is_ordinary_partial {n : ℕ} (hn : 4 ≤ n)
(i : Fin (n-4)) (p : MvPolynomial (Fin n) ℂ) :
localizedPartial (extendedLocalizationEquiv hn) (blocksToOutput hn (Sum.inl i))
(extendedLocMap hn p) =
extendedLocMap hn (pderiv (blocksToAmbient hn (Sum.inl i)) p) := by
apply (extendedLocalizationEquiv hn).injective
rw [localizedPartial_apply, AlgEquiv.apply_symm_apply, extendedLocalizationEquiv_extra_pderiv, extendedLocMap_extra_pderiv]
end AbhyankarSathaye.CommutingDerivations
end
end OAI
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