| import OAI.NumberTheory.PiExponent.LocalAlgebra.PrimeNormalRigidity |
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| namespace OAI |
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| noncomputable section |
| namespace PiExponent |
| open Module MvPolynomial NormalBasisRigidity |
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| variable {C ι : Type*} [Field C] [CharZero C] [IsAlgClosed C] |
| [Fintype ι] [LinearOrder ι] |
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| omit [LinearOrder ι] in |
| theorem coordinate_constant_of_unique_normal_basis |
| (Q : Ideal (MvPolynomial ι C)) [Q.IsPrime] |
| (hq : (polynomialTangent (polynomialResidueMap Q) Q).mkQ ≠ 0) |
| (hunique : ∀ A B : Finset ι, |
| IsNormalBasis (K := PolynomialResidueField Q) |
| (fun j => (polynomialTangent (polynomialResidueMap Q) Q).mkQ |
| (Pi.basisFun (PolynomialResidueField Q) ι j)) A → |
| IsNormalBasis (K := PolynomialResidueField Q) |
| (fun j => (polynomialTangent (polynomialResidueMap Q) Q).mkQ |
| (Pi.basisFun (PolynomialResidueField Q) ι j)) B → A = B) : |
| ∃ i : ι, ∃ c : C, X i - MvPolynomial.C c ∈ Q := by |
| let φ := polynomialResidueMap Q |
| let q := (polynomialTangent φ Q).mkQ |
| let b := Pi.basisFun (PolynomialResidueField Q) ι |
| have hex : ∃ i, q (b i) ≠ 0 := by |
| by_contra! h |
| apply hq |
| exact b.ext h |
| obtain ⟨i, hi⟩ := hex |
| obtain ⟨A, hiA, hA⟩ := exists_normalBasis_containing (fun j => q (b j)) |
| (polynomialNormal_span φ Q b) i hi |
| have hmand : ∀ B, IsNormalBasis (K := PolynomialResidueField Q) (fun j => q (b j)) B → i ∈ B := by |
| intro B hB |
| rw [← hunique A B hA hB] |
| exact hiA |
| have htangent : ∀ t ∈ polynomialTangent φ Q, t i = 0 := by |
| intro t ht |
| have hh := mandatory_coordinate_eq_zero b q (polynomialNormal_span φ Q b) |
| i hmand t ((Submodule.Quotient.mk_eq_zero _).mpr ht) |
| simpa only [b, Pi.basisFun_repr] using hh |
| have hD := differential_eq_zero_of_polynomialTangent φ Q |
| (polynomialResidueMap_eq_zero Q) i htangent |
| have : Algebra.EssFiniteType C (PolynomialResidueField Q) := |
| polynomialResidueField_essFiniteType Q |
| obtain ⟨c, hc⟩ := constant_of_differential_eq_zero (φ (X i)) hD |
| exact ⟨i, c, coordinate_sub_constant_mem_prime Q i c hc⟩ |
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| theorem coordinate_constant_of_normal_discrepancy_excluded |
| (Q : Ideal (MvPolynomial ι C)) [Q.IsPrime] |
| (hq : (polynomialTangent (polynomialResidueMap Q) Q).mkQ ≠ 0) |
| (hexclude : ∀ A B : Finset ι, |
| IsNormalBasis (K := PolynomialResidueField Q) |
| (fun j => (polynomialTangent (polynomialResidueMap Q) Q).mkQ |
| (Pi.basisFun (PolynomialResidueField Q) ι j)) A → |
| IsNormalBasis (K := PolynomialResidueField Q) |
| (fun j => (polynomialTangent (polynomialResidueMap Q) Q).mkQ |
| (Pi.basisFun (PolynomialResidueField Q) ι j)) B → |
| ∀ i, i ∈ A → i ∉ B → (∀ j, i < j → (j ∈ A ↔ j ∈ B)) → False) : |
| ∃ i : ι, ∃ c : C, X i - MvPolynomial.C c ∈ Q := by |
| apply coordinate_constant_of_unique_normal_basis Q hq |
| exact normalBasis_unique_of_discrepancy_excluded _ hexclude |
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| omit [CharZero C] [IsAlgClosed C] [Fintype ι] [LinearOrder ι] in |
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| theorem coordinate_constant_at_most_one_center {J : Type*} |
| (Q : Ideal (MvPolynomial ι C)) (centers : J → ι → C) |
| (i : ι) (c : C) (hc : X i - MvPolynomial.C c ∈ Q) |
| (hinj : Function.Injective (fun j => centers j i)) |
| (j k : J) |
| (hj : ∀ p ∈ Q, MvPolynomial.eval (centers j) p = 0) |
| (hk : ∀ p ∈ Q, MvPolynomial.eval (centers k) p = 0) : j = k := by |
| have hjc := hj _ hc |
| have hkc := hk _ hc |
| simp only [eval_sub, eval_X, eval_C, sub_eq_zero] at hjc hkc |
| exact hinj (hjc.trans hkc.symm) |
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| end PiExponent |
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| end |
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| end OAI |
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