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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 | import OAI.NumberTheory.PiExponent.Jets.CompactLogJetIdeal
namespace OAI
noncomputable section
open scoped BigOperators
namespace PiExponent.JetProductLocalization
open CompactJetPolynomial LogarithmicContactIdeal
variable {A : Type*} [CommRing A] [IsLocalRing A] [Algebra ℂ A]
[Algebra.IsIntegral ℂ (IsLocalRing.ResidueField A)]
def residueAlgHom : A →ₐ[ℂ] ℂ where
__ := CurveLocalOrder.residueAugmentation ℂ A
commutes' := CurveLocalOrder.residueAugmentation_algebraMap ℂ A
theorem residue_ne_zero_iff_isUnit (a : A) :
CurveLocalOrder.residueAugmentation ℂ A a ≠ 0 ↔ IsUnit a := by
change (CurveLocalOrder.residueCoefficientEquiv ℂ A).symm
((IsLocalRing.residue A) a) ≠ 0 ↔ IsUnit a
rw [map_ne_zero_iff _ (CurveLocalOrder.residueCoefficientEquiv ℂ A).symm.injective,
IsLocalRing.residue_ne_zero_iff_isUnit]
theorem residue_polynomial_aeval (a : A) (P : Polynomial ℂ) :
CurveLocalOrder.residueAugmentation ℂ A (Polynomial.aeval a P) =
P.eval (CurveLocalOrder.residueAugmentation ℂ A a) := by
simpa [Polynomial.coe_aeval_eq_eval, residueAlgHom] using
(Polynomial.aeval_algHom_apply (residueAlgHom (A := A)) a P).symm
theorem residue_truncatedCoordinates_succ {m : ℕ}
(c : Fin m → ℂ) (T : Fin m → ℕ) (y : A) (x : Fin m → A)
(hy : CurveLocalOrder.residueAugmentation ℂ A y = 1) (i : Fin m) :
CurveLocalOrder.residueAugmentation ℂ A (truncatedCoordinates c y x T i.succ) =
CurveLocalOrder.residueAugmentation ℂ A (x i) - c i := by
have hlog : CurveLocalOrder.residueAugmentation ℂ A
(Polynomial.aeval (y-1) (PowerSeries.trunc (T i) (PowerSeries.log ℂ))) = 0 := by
rw [residue_polynomial_aeval, map_sub, map_one, hy, sub_self]
exact logPolynomials_eval_zero T i
simp only [truncatedCoordinates, Fin.cases_succ, map_sub,
CurveLocalOrder.residueAugmentation_algebraMap, hlog, sub_zero]
theorem logarithmicIdeal_eq_top_of_residue_ne {m : ℕ}
(c : Fin m → ℂ) (T : Fin m → ℕ) (e : Fin (m+1) → ℕ)
(y : A) (x : Fin m → A)
(hne : ¬ (CurveLocalOrder.residueAugmentation ℂ A y = 1 ∧
∀ i, CurveLocalOrder.residueAugmentation ℂ A (x i) = c i)) :
logarithmicIdeal c y x T e = ⊤ := by
classical
have hunit : ∃ i : Fin (m+1), IsUnit (truncatedCoordinates c y x T i) := by
by_cases hy : CurveLocalOrder.residueAugmentation ℂ A y = 1
· have hx : ¬ ∀ i, CurveLocalOrder.residueAugmentation ℂ A (x i) = c i :=
fun hx => hne ⟨hy, hx⟩
obtain ⟨i, hi⟩ := not_forall.mp hx
refine ⟨i.succ, (residue_ne_zero_iff_isUnit _).mp ?_⟩
rw [residue_truncatedCoordinates_succ c T y x hy i]
exact sub_ne_zero.mpr hi
· refine ⟨0, (residue_ne_zero_iff_isUnit _).mp ?_⟩
change CurveLocalOrder.residueAugmentation ℂ A (y-1) ≠ 0
rw [map_sub, map_one]
exact sub_ne_zero.mpr hy
obtain ⟨i, hi⟩ := hunit
apply Ideal.eq_top_of_isUnit_mem _
(show truncatedCoordinates c y x T i ^ e i ∈ logarithmicIdeal c y x T e from
Ideal.subset_span (Set.mem_range_self i))
exact hi.pow (e i)
theorem map_powerIdeal_eq_top_of_ne {J : Type*} {m : ℕ}
(c : J → Fin m → ℂ) (hc : Function.Injective c)
(T : Fin m → ℕ) (e : Fin (m+1) → ℕ) (y : A) (x : Fin m → A)
(j k : J) (hkj : k ≠ j)
(hy : CurveLocalOrder.residueAugmentation ℂ A y = 1)
(hx : ∀ i, CurveLocalOrder.residueAugmentation ℂ A (x i) = c j i) :
(powerIdeal (c k) (logPolynomials T) e).map
(MvPolynomial.aeval (Fin.cases y x)).toRingHom = ⊤ := by
rw [map_powerIdeal_aeval]
apply logarithmicIdeal_eq_top_of_residue_ne
rw [hy]
rintro ⟨_, hk⟩
apply hkj
apply hc
funext i
exact (hk i).symm.trans (hx i)
omit [IsLocalRing A] [Algebra.IsIntegral ℂ (IsLocalRing.ResidueField A)] in
theorem map_polynomialIdeal {J : Type*} [Fintype J] {m : ℕ}
(c : J → Fin m → ℂ) (T : Fin m → ℕ) (e : Fin (m+1) → ℕ)
(y : A) (x : Fin m → A) :
(CompactLogJetIdeal.polynomialIdeal c T e).map
(MvPolynomial.aeval (Fin.cases y x)).toRingHom =
∏ j, logarithmicIdeal (c j) y x T e := by
unfold CompactLogJetIdeal.polynomialIdeal
change Ideal.mapHom (MvPolynomial.aeval (Fin.cases y x)).toRingHom _ = _
rw [map_prod]
apply Finset.prod_congr rfl
intro j _
exact map_powerIdeal_aeval (c j) T e y x
theorem map_polynomialIdeal_eq_selected {J : Type*} [Fintype J] {m : ℕ}
(c : J → Fin m → ℂ) (hc : Function.Injective c)
(T : Fin m → ℕ) (e : Fin (m+1) → ℕ) (y : A) (x : Fin m → A) (j : J)
(hy : CurveLocalOrder.residueAugmentation ℂ A y = 1)
(hx : ∀ i, CurveLocalOrder.residueAugmentation ℂ A (x i) = c j i) :
(CompactLogJetIdeal.polynomialIdeal c T e).map
(MvPolynomial.aeval (Fin.cases y x)).toRingHom = logarithmicIdeal (c j) y x T e := by
classical
rw [map_polynomialIdeal]
apply Finset.prod_eq_single j
· intro k _ hkj
rw [Ideal.one_eq_top]
rw [← map_powerIdeal_aeval]
exact map_powerIdeal_eq_top_of_ne c hc T e y x j k hkj hy hx
· simp
theorem map_polynomialIdeal_eq_top_outside {J : Type*} [Fintype J] {m : ℕ}
(c : J → Fin m → ℂ) (T : Fin m → ℕ) (e : Fin (m+1) → ℕ)
(y : A) (x : Fin m → A)
(hout : ¬ ∃ j, CurveLocalOrder.residueAugmentation ℂ A y = 1 ∧
∀ i, CurveLocalOrder.residueAugmentation ℂ A (x i) = c j i) :
(CompactLogJetIdeal.polynomialIdeal c T e).map
(MvPolynomial.aeval (Fin.cases y x)).toRingHom = ⊤ := by
rw [map_polynomialIdeal]
rw [← Ideal.one_eq_top]
apply Finset.prod_eq_one
intro j _
rw [Ideal.one_eq_top]
exact logarithmicIdeal_eq_top_of_residue_ne (c j) T e y x (fun h => hout ⟨j, h⟩)
end PiExponent.JetProductLocalization
end
end OAI
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