openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /Polynomials /PolynomialLocalResidueResolution.lean
| 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 | |