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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 | import OAI.AlgebraicGeometry.CommutingDerivations.ExtendedLocalization
import OAI.AlgebraicGeometry.CommutingDerivations.ConcreteKernelIntersection
namespace OAI
/-!
The coefficient algebra of the localization intersects the full ambient
ring in exactly `ℂ[Fₙ]` in every dimension `n ≥ 4`.
-/
noncomputable section
namespace AbhyankarSathaye.CommutingDerivations
open MvPolynomial
@[simp] theorem extendedCoefficientMap_complex (n : ℕ) (a : ℂ) :
extendedCoefficientMap n (algebraMap ℂ LaurentCoefficients a) =
algebraMap ℂ (ExtendedLocalizedAmbient n) a := by
change C (coefficientMap (algebraMap ℂ LaurentCoefficients a)) = _
rw [coefficientMap_complex_intersection]
rfl
theorem extendedCoefficientMap_polynomial {n : ℕ} (hn : 4 ≤ n) (q : Polynomial ℂ) :
extendedCoefficientMap n (algebraMap (Polynomial ℂ) LaurentCoefficients q) =
extendedLocMap hn (Polynomial.aeval (extendedAmbientParameter hn) q) := by
change C (coefficientMap (algebraMap (Polynomial ℂ) LaurentCoefficients q)) = _
rw [coefficientMap_polynomial]
let original : R →ₐ[ℂ] ExtendedLocalizedAmbient n :=
(IsScalarTower.toAlgHom ℂ LocalizedAmbient (ExtendedLocalizedAmbient n)).comp
(IsScalarTower.toAlgHom ℂ R LocalizedAmbient)
have he : original.comp (Polynomial.aeval ambientC) =
(extendedLocAlgHom hn).comp (Polynomial.aeval (extendedAmbientParameter hn)) := by
apply Polynomial.algHom_ext
simp only [AlgHom.comp_apply, Polynomial.aeval_X]
change C (loc ambientC) = extendedLocMap hn (extendedF hn + 1)
exact (extendedLocMap_c hn).symm
exact AlgHom.congr_fun he q
theorem extendedCoefficientMap_parameter {n : ℕ} (hn : 4 ≤ n) :
extendedCoefficientMap n coefficientC = extendedLocMap hn (extendedAmbientParameter hn) := by
change C (coefficientMap coefficientC) = extendedLocMap hn (extendedF hn + 1)
rw [coefficientMap_C, extendedLocMap_c]
/-- Exact input to the generic commuting-family construction, specialized to
the actual all-dimensional localization and coordinate equivalence. -/
theorem extended_localization_kernel_intersection {n : ℕ} (hn : 4 ≤ n)
(r : MvPolynomial (Fin n) ℂ) :
(∃ b : LaurentCoefficients,
extendedLocalizationEquiv hn (extendedLocMap hn r) = C b) ↔
r ∈ Algebra.adjoin ℂ ({extendedF hn} : Set (MvPolynomial (Fin n) ℂ)) := by
let := extendedAlgebra hn
let : IsLocalization.Away (extendedAmbientParameter hn) (ExtendedLocalizedAmbient n) := by
change IsLocalization.Away (extendedF hn + 1) (ExtendedLocalizedAmbient n)
exact extended_isLocalization hn
constructor
· rintro ⟨b, hb⟩
have hpre : extendedLocMap hn r = extendedCoefficientMap n b := by
apply (extendedLocalizationEquiv hn).injective
rw [hb, extendedLocalizationEquiv_coefficient]
obtain ⟨m, q, hq⟩ := IsLocalization.Away.surj (Polynomial.X : Polynomial ℂ) b
change b * coefficientC^m = algebraMap (Polynomial ℂ) LaurentCoefficients q at hq
have he : extendedCoefficientMap n b *
(extendedLocMap hn (extendedAmbientParameter hn))^m =
extendedLocMap hn (Polynomial.aeval (extendedAmbientParameter hn) q) := by
simpa only [map_mul, map_pow, extendedCoefficientMap_parameter hn, extendedCoefficientMap_polynomial hn] using congrArg (extendedCoefficientMap n) hq
apply (extendedParameter_laurent_intersection hn (ExtendedLocalizedAmbient n) r).mp
refine ⟨m, q, ?_⟩
change (extendedLocMap hn (extendedAmbientParameter hn))^m * extendedLocMap hn r =
extendedLocMap hn (Polynomial.aeval (extendedAmbientParameter hn) q)
rw [hpre, mul_comm]
exact he
· intro hr
rw [Algebra.adjoin_singleton_eq_range_aeval] at hr
obtain ⟨q, hq⟩ := hr
obtain ⟨p, hp⟩ := (exists_aeval_add_one_iff (K := ℂ) (extendedF hn) r).mpr ⟨q, hq.symm⟩
refine ⟨algebraMap (Polynomial ℂ) LaurentCoefficients p, ?_⟩
calc
extendedLocalizationEquiv hn (extendedLocMap hn r) =
extendedLocalizationEquiv hn
(extendedCoefficientMap n (algebraMap (Polynomial ℂ) LaurentCoefficients p)) := by
apply congrArg (extendedLocalizationEquiv hn)
rw [extendedCoefficientMap_polynomial]
exact congrArg (extendedLocMap hn) hp
_ = C (algebraMap (Polynomial ℂ) LaurentCoefficients p) :=
extendedLocalizationEquiv_coefficient hn _
end AbhyankarSathaye.CommutingDerivations
end
end OAI
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