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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 | import OAI.AlgebraicGeometry.CommutingDerivations.CoordinatePartials
namespace OAI
/-! Transport the actual coordinate partials through a specified, proved
algebra equivalence. Coefficients in the whole relative coefficient algebra
are fixed, even when the derivations are viewed over a smaller ground ring. -/
noncomputable section
namespace AbhyankarSathaye.CommutingDerivations
open MvPolynomial
variable {K B L I : Type*} [CommRing K] [CommRing B] [CommRing L]
[Algebra K B] [Algebra K L]
def localizedPartial (e : L ≃ₐ[K] MvPolynomial I B) (i : I) : Derivation K L L :=
transportDerivation e ((pderiv i : Derivation B (MvPolynomial I B) _).restrictScalars K)
@[simp] theorem localizedPartial_apply (e : L ≃ₐ[K] MvPolynomial I B) (i : I) (a : L) :
localizedPartial e i a = e.symm (pderiv i (e a)) := rfl
theorem localizedPartial_locallyNilpotent (e : L ≃ₐ[K] MvPolynomial I B) (i : I) :
LocallyNilpotent (localizedPartial e i) :=
transportDerivation_locallyNilpotent e _ (coordinate_partial_locallyNilpotent i)
theorem localizedPartials_commute (e : L ≃ₐ[K] MvPolynomial I B) (i j : I) (a : L) :
localizedPartial e i (localizedPartial e j a) =
localizedPartial e j (localizedPartial e i a) := by
simp only [localizedPartial_apply, e.apply_symm_apply, coordinate_partials_commute i j]
theorem localizedPartial_fixes_relative (e : L ≃ₐ[K] MvPolynomial I B)
(i : I) (a : L) (b : B) (hab : e a = C b) : localizedPartial e i a = 0 := by
rw [localizedPartial_apply, hab, pderiv_C, map_zero]
theorem localizedPartial_coordinate [DecidableEq I]
(e : L ≃ₐ[K] MvPolynomial I B) (i j : I) :
localizedPartial e i (e.symm (X j)) = if i = j then 1 else 0 := by
simp [localizedPartial_apply, pderiv_X, Pi.single_apply, eq_comm]
theorem localizedPartial_kernel (e : L ≃ₐ[K] MvPolynomial I B) (i : I) (a : L) :
localizedPartial e i a = 0 ↔ pderiv i (e a) = 0 := by
constructor
· intro h
have hh := congrArg e h
simpa only [localizedPartial_apply, e.apply_symm_apply, map_zero] using hh
· intro h
rw [localizedPartial_apply, h, map_zero]
theorem localizedPartials_common_kernel [IsDomain B] [CharZero B]
(e : L ≃ₐ[K] MvPolynomial I B) (a : L) :
(∀ i, localizedPartial e i a = 0) ↔ ∃ b : B, e a = C b := by
simp only [localizedPartial_kernel]
exact coordinate_partials_common_kernel (e a)
end AbhyankarSathaye.CommutingDerivations
end
end OAI
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