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a72760f | 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 | import OAI.NumberTheory.PiExponent.LocalAlgebra.PrimeNormalRigidity
namespace OAI
noncomputable section
namespace PiExponent
open Module MvPolynomial NormalBasisRigidity
variable {C ι : Type*} [Field C] [CharZero C] [IsAlgClosed C]
[Fintype ι] [LinearOrder ι]
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⟩
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
omit [CharZero C] [IsAlgClosed C] [Fintype ι] [LinearOrder ι] in
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)
end PiExponent
end
end OAI
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