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b797a5b | 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 | import Mathlib.FieldTheory.IntermediateField.Adjoin.Basic
import Mathlib.FieldTheory.SeparablyGenerated
import Mathlib.RingTheory.LocalRing.ResidueField.Instances
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Smooth.Basic
namespace OAI
noncomputable section
namespace PiExponentJets.PolynomialLocalResidueResolution
section LocalCoefficientField
variable {K R ι : Type*} [Field K] [CommRing R] [IsLocalRing R] [Algebra K R]
def residueAlgHom : R →ₐ[K] IsLocalRing.ResidueField R :=
{ IsLocalRing.residue R with commutes' := fun _ => rfl }
theorem isUnit_aeval_of_residue_independent (x : ι → R)
(hx : AlgebraicIndependent K (fun i => IsLocalRing.residue R (x i)))
(f : MvPolynomial ι K) (hf : f ≠ 0) : IsUnit (MvPolynomial.aeval x f) := by
apply (IsLocalRing.residue_ne_zero_iff_isUnit (MvPolynomial.aeval x f)).mp
intro hz
apply hf
apply hx.eq_zero_of_aeval_eq_zero
have he := MvPolynomial.comp_aeval_apply (f := x) (residueAlgHom (K := K) (R := R)) f
exact he.symm.trans hz
def rationalCoefficientMap (x : ι → R)
(hx : AlgebraicIndependent K (fun i => IsLocalRing.residue R (x i))) :
FractionRing (MvPolynomial ι K) →+* R :=
IsLocalization.lift (M := nonZeroDivisors (MvPolynomial ι K))
(S := FractionRing (MvPolynomial ι K)) (g := (MvPolynomial.aeval x).toRingHom)
(fun f => isUnit_aeval_of_residue_independent x hx f (nonZeroDivisors.coe_ne_zero f))
@[simp] theorem rationalCoefficientMap_polynomial (x : ι → R)
(hx : AlgebraicIndependent K (fun i => IsLocalRing.residue R (x i)))
(f : MvPolynomial ι K) :
rationalCoefficientMap x hx
(algebraMap (MvPolynomial ι K) (FractionRing (MvPolynomial ι K)) f) =
MvPolynomial.aeval x f :=
IsLocalization.lift_eq _ _
theorem rationalCoefficientMap_injective (x : ι → R)
(hx : AlgebraicIndependent K (fun i => IsLocalRing.residue R (x i))) :
Function.Injective (rationalCoefficientMap x hx) :=
(rationalCoefficientMap x hx).injective
theorem residue_rationalCoefficientMap_injective (x : ι → R)
(hx : AlgebraicIndependent K (fun i => IsLocalRing.residue R (x i))) :
Function.Injective ((IsLocalRing.residue R).comp (rationalCoefficientMap x hx)) :=
((IsLocalRing.residue R).comp (rationalCoefficientMap x hx)).injective
end LocalCoefficientField
variable (K : Type*) [Field K] (n : ℕ)
variable (Q : Ideal (MvPolynomial (Fin n) K)) [Q.IsPrime]
abbrev LocalPolynomialRing := Localization.AtPrime Q
abbrev LocalPolynomialResidue := Q.ResidueField
theorem actual_residue_field : LocalPolynomialResidue K n Q =
IsLocalRing.ResidueField (LocalPolynomialRing K n Q) := rfl
def residueCoordinate (i : Fin n) : LocalPolynomialResidue K n Q :=
algebraMap (MvPolynomial (Fin n) K) (LocalPolynomialResidue K n Q) (MvPolynomial.X i)
theorem localPolynomial_essFiniteType :
Algebra.EssFiniteType K (LocalPolynomialRing K n Q) := by infer_instance
theorem localResidue_essFiniteType :
Algebra.EssFiniteType K (LocalPolynomialResidue K n Q) := by
exact Algebra.EssFiniteType.comp K (MvPolynomial (Fin n) K) _
theorem exists_separating_residue_basis [PerfectField K] :
∃ s : Finset (LocalPolynomialResidue K n Q),
IsTranscendenceBasis K ((↑) : s → LocalPolynomialResidue K n Q) ∧
Algebra.IsSeparable (IntermediateField.adjoin K (s : Set (LocalPolynomialResidue K n Q)))
(LocalPolynomialResidue K n Q) := by
let := localResidue_essFiniteType K n Q
exact exists_isTranscendenceBasis_and_isSeparable_of_perfectField K _
theorem exists_local_lifts (s : Finset (LocalPolynomialResidue K n Q)) :
∃ x : s → LocalPolynomialRing K n Q,
∀ i, IsLocalRing.residue (LocalPolynomialRing K n Q) (x i) = (i : LocalPolynomialResidue K n Q) := by
classical
choose x hx using fun i : s =>
IsLocalRing.residue_surjective (R := LocalPolynomialRing K n Q) (i : LocalPolynomialResidue K n Q)
exact ⟨x, hx⟩
theorem residue_coordinate_equations
(s : Finset (LocalPolynomialResidue K n Q))
[Algebra.IsSeparable
(IntermediateField.adjoin K (s : Set (LocalPolynomialResidue K n Q)))
(LocalPolynomialResidue K n Q)] (i : Fin n) :
let F := IntermediateField.adjoin K (s : Set (LocalPolynomialResidue K n Q))
let a := residueCoordinate K n Q i
(minpoly F a).Monic ∧ Polynomial.aeval a (minpoly F a) = 0 ∧
Polynomial.aeval a (Polynomial.derivative (minpoly F a)) ≠ 0 := by
dsimp only
refine ⟨minpoly.monic (Algebra.IsSeparable.isIntegral _ _), minpoly.aeval _ _, ?_⟩
exact (Algebra.IsSeparable.isSeparable _ _).aeval_derivative_ne_zero (minpoly.aeval _ _)
end PiExponentJets.PolynomialLocalResidueResolution
end
end OAI
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