File size: 3,308 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 72 73 74 75 | import OAI.AlgebraicGeometry.CommutingDerivations.LocalizationSetup
import OAI.AlgebraicGeometry.CommutingDerivations.ExtendedParameterRetraction
namespace OAI
/-!
The Laurent coefficient-map image intersects the ambient ring in exactly
`ℂ[F]`, by the explicit parameter retraction.
-/
noncomputable section
namespace AbhyankarSathaye.CommutingDerivations
theorem polynomialCoefficientMap_apply (q : Polynomial ℂ) :
polynomialCoefficientMap q = loc (Polynomial.aeval ambientC q) := by
have he := Polynomial.aeval_algHom
(IsScalarTower.toAlgHom ℂ R LocalizedAmbient) ambientC
exact AlgHom.congr_fun he q
@[simp] theorem coefficientMap_polynomial (q : Polynomial ℂ) :
coefficientMap (algebraMap (Polynomial ℂ) LaurentCoefficients q) =
loc (Polynomial.aeval ambientC q) := by
rw [← polynomialCoefficientMap_apply]
unfold coefficientMap
exact IsLocalization.Away.lift_eq _ _ _
/-- Constants in the actual Laurent coefficient algebra have the prescribed
complex scalar image in the actual ambient localization. -/
@[simp] theorem coefficientMap_complex_intersection (a : ℂ) :
coefficientMap (algebraMap ℂ LaurentCoefficients a) = algebraMap ℂ LocalizedAmbient a := by
rw [IsScalarTower.algebraMap_apply ℂ (Polynomial ℂ) LaurentCoefficients]
rw [coefficientMap_polynomial]
simp [loc, IsScalarTower.algebraMap_apply ℂ R LocalizedAmbient]
/-- Literal common-kernel intersection form: membership in the image of the
actual coefficient map is equivalent to an ordinary polynomial in F. -/
theorem coefficientMap_intersection (r : R) :
(∃ a : LaurentCoefficients, loc r = coefficientMap a) ↔
∃ q : Polynomial ℂ, r = Polynomial.aeval F q := by
let : IsLocalization.Away ambientParameter LocalizedAmbient := by
change IsLocalization.Away ambientC LocalizedAmbient
infer_instance
constructor
· rintro ⟨a, ha⟩
obtain ⟨m, q, hq⟩ := IsLocalization.Away.surj (Polynomial.X : Polynomial ℂ) a
change a * coefficientC^m = algebraMap (Polynomial ℂ) LaurentCoefficients q at hq
have he : coefficientMap a * (loc ambientC)^m = loc (Polynomial.aeval ambientC q) := by
simpa only [map_mul, map_pow, coefficientMap_C, coefficientMap_polynomial] using
congrArg coefficientMap hq
have hpoly : ∃ q : Polynomial ℂ, r = Polynomial.aeval ambientParameter q := by
apply (parameter_laurent_intersection LocalizedAmbient r).mp
refine ⟨m, q, ?_⟩
change (loc ambientC)^m * loc r = loc (Polynomial.aeval ambientC q)
rw [ha, mul_comm]
exact he
exact (exists_aeval_add_one_iff (K := ℂ) F r).mp hpoly
· intro hr
obtain ⟨q, hq⟩ := (exists_aeval_add_one_iff (K := ℂ) F r).mpr hr
refine ⟨algebraMap (Polynomial ℂ) LaurentCoefficients q, ?_⟩
rw [coefficientMap_polynomial]
exact congrArg loc hq
theorem coefficientMap_intersection_adjoin (r : R) :
(∃ a : LaurentCoefficients, loc r = coefficientMap a) ↔
r ∈ Algebra.adjoin ℂ ({F} : Set R) := by
rw [coefficientMap_intersection, Algebra.adjoin_singleton_eq_range_aeval]
change (∃ q : Polynomial ℂ, r = Polynomial.aeval F q) ↔
∃ q : Polynomial ℂ, Polynomial.aeval F q = r
simp only [eq_comm]
end AbhyankarSathaye.CommutingDerivations
end
end OAI
|