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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 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 | import OAI.AlgebraicGeometry.CommutingDerivations.LocalizationEquivalence
import OAI.AlgebraicGeometry.AbhyankarSathaye.Stabilization
import Mathlib.RingTheory.MvPolynomial.Localization
namespace OAI
/-!
Localization coordinates in every dimension `n ≥ 4`.
The localization at `extendedF hn + 1` is a polynomial extension of the
four-dimensional localization, with an explicit structural map from the ambient ring.
-/
noncomputable section
namespace AbhyankarSathaye.CommutingDerivations
open MvPolynomial
abbrev ExtendedLocalizedAmbient (n : ℕ) := MvPolynomial (Fin (n-4)) LocalizedAmbient
def extendedLocAlgHom {n : ℕ} (hn : 4 ≤ n) :
MvPolynomial (Fin n) ℂ →ₐ[ℂ] ExtendedLocalizedAmbient n :=
(MvPolynomial.mapAlgHom (σ := Fin (n-4))
(IsScalarTower.toAlgHom ℂ R LocalizedAmbient)).comp (splitAmbient hn).toAlgHom
def extendedLocMap {n : ℕ} (hn : 4 ≤ n) :
MvPolynomial (Fin n) ℂ →+* ExtendedLocalizedAmbient n :=
(extendedLocAlgHom hn).toRingHom
@[instance_reducible]
def extendedAlgebra {n : ℕ} (hn : 4 ≤ n) :
Algebra (MvPolynomial (Fin n) ℂ) (ExtendedLocalizedAmbient n) :=
(extendedLocMap hn).toAlgebra
theorem extendedScalarTower {n : ℕ} (hn : 4 ≤ n) :
letI := extendedAlgebra hn
IsScalarTower ℂ (MvPolynomial (Fin n) ℂ) (ExtendedLocalizedAmbient n) :=
IsScalarTower.of_algHom (extendedLocAlgHom hn)
@[simp] theorem extendedLocMap_c {n : ℕ} (hn : 4 ≤ n) :
extendedLocMap hn (extendedF hn+1) = C (loc ambientC) := by
simp [extendedLocMap, extendedLocAlgHom, splitAmbient_F, ambientC, loc]
change MvPolynomial.map (algebraMap R LocalizedAmbient) (C F) =
C ((algebraMap R LocalizedAmbient) F)
exact map_C _ _
/-- This polynomial extension is the localization of the exact full ambient
ring at the powers of extendedF+1, with the displayed structural map. -/
theorem extended_isLocalization {n : ℕ} (hn : 4 ≤ n) :
letI := extendedAlgebra hn
IsLocalization.Away (extendedF hn+1) (ExtendedLocalizedAmbient n) := by
let := extendedAlgebra hn
let : Algebra (MvPolynomial (Fin (n-4)) R) (ExtendedLocalizedAmbient n) :=
MvPolynomial.algebraMvPolynomial
have : IsLocalization ((Submonoid.powers ambientC).map
(C : R →+* MvPolynomial (Fin (n-4)) R)) (ExtendedLocalizedAmbient n) :=
MvPolynomial.isLocalization _ _
apply IsLocalization.of_ringEquiv_left (splitAmbient hn).toRingEquiv
(M₁ := (Submonoid.powers ambientC).map (C : R →+* MvPolynomial (Fin (n-4)) R))
· simp [Submonoid.map_powers, splitAmbient_F, ambientC]
· intro f
rfl
def extendedCoefficientMap (n : ℕ) : LaurentCoefficients →+* ExtendedLocalizedAmbient n :=
C.comp coefficientMap
/-- Complex-linear localization coordinates, with exactly n−1 independent
polynomial variables and the original Laurent coefficient ring. -/
def extendedLocalizationEquiv {n : ℕ} (hn : 4 ≤ n) :
ExtendedLocalizedAmbient n ≃ₐ[ℂ] MvPolynomial (Fin (n-1)) LaurentCoefficients :=
(MvPolynomial.mapAlgEquiv (Fin (n-4)) localizationComplexEquiv).trans
(((sumAlgEquiv LaurentCoefficients (Fin (n-4)) (Fin 3)).symm.restrictScalars ℂ).trans
((renameEquiv LaurentCoefficients (blocksToOutput hn)).restrictScalars ℂ))
@[simp] theorem extendedLocalizationEquiv_coefficient {n : ℕ} (hn : 4 ≤ n)
(a : LaurentCoefficients) :
extendedLocalizationEquiv hn (extendedCoefficientMap n a) = C a := by
have ha : localizationComplexEquiv (coefficientMap a) = C a :=
relativeForward_coefficient a
change rename (blocksToOutput hn)
((sumAlgEquiv LaurentCoefficients (Fin (n-4)) (Fin 3)).symm
(MvPolynomial.map localizationComplexEquiv (C (coefficientMap a)))) = C a
rw [map_C]
change rename (blocksToOutput hn)
((sumAlgEquiv LaurentCoefficients (Fin (n-4)) (Fin 3)).symm
(C (localizationComplexEquiv (coefficientMap a)))) = C a
rw [ha, sumAlgEquiv_symm_C_C, rename_C]
@[simp] theorem extendedLocalizationEquiv_c {n : ℕ} (hn : 4 ≤ n) :
extendedLocalizationEquiv hn (extendedLocMap hn (extendedF hn+1)) = C coefficientC := by
rw [extendedLocMap_c, ← coefficientMap_C]
exact extendedLocalizationEquiv_coefficient hn coefficientC
@[instance_reducible]
def extendedRelativeAlgebra (n : ℕ) :
Algebra LaurentCoefficients (ExtendedLocalizedAmbient n) :=
(extendedCoefficientMap n).toAlgebra
/-- The all-dimensional equivalence fixes precisely the specified Laurent
coefficient map, as an actual relative algebra equivalence. -/
def extendedRelativeLocalizationEquiv {n : ℕ} (hn : 4 ≤ n) :
letI := extendedRelativeAlgebra n
ExtendedLocalizedAmbient n ≃ₐ[LaurentCoefficients]
MvPolynomial (Fin (n-1)) LaurentCoefficients :=
letI := extendedRelativeAlgebra n
{ (extendedLocalizationEquiv hn).toRingEquiv with
commutes' := extendedLocalizationEquiv_coefficient hn }
end AbhyankarSathaye.CommutingDerivations
end
end OAI
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