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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 | import Mathlib
import OAI.NumberTheory.PiExponent.Geometry.WeightedCurveDegree
import OAI.NumberTheory.PiExponent.Jets.IntrinsicLength
namespace OAI
open scoped BigOperators nonZeroDivisors
open IsLocalRing IsDiscreteValuationRing
namespace PiExponent.CurveLocalOrder
theorem length_quotient_span_eq_addVal
{A : Type*} [CommRing A] [IsDomain A] [IsDiscreteValuationRing A]
{a : A} (ha : a ≠ 0) :
Module.length A (A ⧸ Ideal.span {a}) = IsDiscreteValuationRing.addVal A a := by
obtain ⟨ϖ, hϖ⟩ := IsDiscreteValuationRing.exists_irreducible A
obtain ⟨n, u, rfl⟩ := IsDiscreteValuationRing.eq_unit_mul_pow_irreducible ha hϖ
rw [IsDiscreteValuationRing.addVal_def' u hϖ]
rw [Ideal.span_singleton_mul_left_unit u.isUnit,
← Ideal.span_singleton_pow, ← hϖ.maximalIdeal_eq]
exact IsDiscreteValuationRing.length_quotient_pow_maximalIdeal A n
def enatToIntegerOrder : ℕ∞ →+ WithTop ℤ where
toFun := ENat.map (fun n : ℕ => (n : ℤ))
map_zero' := by simp
map_add' a b := ENat.map_add (Nat.castAddMonoidHom ℤ) a b
@[simp] theorem enatToIntegerOrder_top : enatToIntegerOrder ⊤ = ⊤ := rfl
@[simp] theorem enatToIntegerOrder_natCast (n : ℕ) :
enatToIntegerOrder n = (n : WithTop ℤ) := rfl
@[simp] theorem enatToIntegerOrder_eq_top_iff (n : ℕ∞) :
enatToIntegerOrder n = ⊤ ↔ n = ⊤ := ENat.map_eq_top_iff
theorem enatToIntegerOrder_monotone : Monotone enatToIntegerOrder := by
apply ENat.monotone_map_iff.mpr
intro m n h
change (m : ℤ) ≤ (n : ℤ)
exact_mod_cast h
noncomputable def localAddValuation
(A : Type*) [CommRing A] [IsDomain A] [IsDiscreteValuationRing A] :
AddValuation A (WithTop ℤ) :=
(IsDiscreteValuationRing.addVal A).map enatToIntegerOrder
enatToIntegerOrder_top enatToIntegerOrder_monotone
@[simp] theorem localAddValuation_apply
{A : Type*} [CommRing A] [IsDomain A] [IsDiscreteValuationRing A] (a : A) :
localAddValuation A a = enatToIntegerOrder (IsDiscreteValuationRing.addVal A a) := rfl
@[simp] theorem localAddValuation_eq_top_iff
{A : Type*} [CommRing A] [IsDomain A] [IsDiscreteValuationRing A] (a : A) :
localAddValuation A a = ⊤ ↔ a = 0 := by
rw [localAddValuation_apply, enatToIntegerOrder_eq_top_iff,
IsDiscreteValuationRing.addVal_eq_top_iff]
theorem nonZeroDivisors_le_primeCompl
(A : Type*) [CommRing A] [IsDomain A] [IsDiscreteValuationRing A] :
nonZeroDivisors A ≤ (localAddValuation A).toValuation.supp.primeCompl := by
intro a ha
change localAddValuation A a ≠ ⊤
rw [ne_eq, localAddValuation_eq_top_iff]
exact mem_nonZeroDivisors_iff_ne_zero.mp ha
noncomputable def fractionAddValuation
(A K : Type*) [CommRing A] [IsDomain A] [IsDiscreteValuationRing A]
[Field K] [Algebra A K] [IsFractionRing A K] : AddValuation K (WithTop ℤ) :=
AddValuation.ofValuation
((localAddValuation A).toValuation.extendToLocalization
(nonZeroDivisors_le_primeCompl A) K)
@[simp] theorem fractionAddValuation_algebraMap
{A K : Type*} [CommRing A] [IsDomain A] [IsDiscreteValuationRing A]
[Field K] [Algebra A K] [IsFractionRing A K] (a : A) :
fractionAddValuation A K (algebraMap A K a) = localAddValuation A a := by
exact Valuation.extendToLocalization_apply_map_apply
(localAddValuation A).toValuation (nonZeroDivisors_le_primeCompl A) K a
theorem fraction_order_eq_principal_colength
{A K : Type*} [CommRing A] [IsDomain A] [IsDiscreteValuationRing A]
[Field K] [Algebra A K] [IsFractionRing A K] {a : A} (ha : a ≠ 0) :
fractionAddValuation A K (algebraMap A K a) =
enatToIntegerOrder (Module.length A (A ⧸ Ideal.span {a})) := by
rw [fractionAddValuation_algebraMap, localAddValuation_apply,
length_quotient_span_eq_addVal ha]
theorem principal_colength_ne_top
{A : Type*} [CommRing A] [IsDomain A] [IsDiscreteValuationRing A]
{a : A} (ha : a ≠ 0) : Module.length A (A ⧸ Ideal.span {a}) ≠ ⊤ := by
rw [length_quotient_span_eq_addVal ha]
exact IsDiscreteValuationRing.addVal_eq_top_iff.not.mpr ha
theorem integerOrder_field_image_eq_length
{A K : Type*} [CommRing A] [IsDomain A] [IsDiscreteValuationRing A]
[Field K] [Algebra A K] [IsFractionRing A K] {a : A} (ha : a ≠ 0) :
WeightedCurveDegree.integerOrder (fractionAddValuation A K)
(Units.mk0 (algebraMap A K a) ((map_ne_zero_iff _ (IsFractionRing.injective A K)).mpr ha)) =
((Module.length A (A ⧸ Ideal.span {a})).toNat : ℤ) := by
apply WithTop.coe_injective
rw [WeightedCurveDegree.coe_integerOrder]
change fractionAddValuation A K (algebraMap A K a) = _
rw [fraction_order_eq_principal_colength ha]
obtain ⟨n, hn⟩ := ENat.ne_top_iff_exists.mp (principal_colength_ne_top ha)
rw [← hn]
simp
theorem atPrime_principal_colength
{S : Type*} [CommRing S] [IsDedekindDomain S]
(q : IsDedekindDomain.HeightOneSpectrum S) {a : S} (ha : a ≠ 0) :
let A := Localization.AtPrime q.asIdeal
letI : IsDiscreteValuationRing A :=
IsLocalization.AtPrime.isDiscreteValuationRing_of_dedekind_domain S q.ne_bot A
Module.length A (A ⧸ Ideal.span {algebraMap S A a}) =
IsDiscreteValuationRing.addVal A (algebraMap S A a) := by
dsimp only
let : IsDiscreteValuationRing (Localization.AtPrime q.asIdeal) :=
IsLocalization.AtPrime.isDiscreteValuationRing_of_dedekind_domain S q.ne_bot _
apply length_quotient_span_eq_addVal
exact (map_ne_zero_iff _ (IsLocalization.injective
(Localization.AtPrime q.asIdeal) q.asIdeal.primeCompl_le_nonZeroDivisors)).mpr ha
theorem ramificationIdx_eq_local_order
{R S : Type*} [CommRing R] [CommRing S] [IsDedekindDomain S]
[Algebra R S] (t : R) (q : Ideal S) [q.IsPrime]
[q.LiesOver (Ideal.span {t})] (hq : q ≠ ⊥)
(ht : algebraMap R (Localization.AtPrime q) t ≠ 0) :
letI : IsDiscreteValuationRing (Localization.AtPrime q) :=
IsLocalization.AtPrime.isDiscreteValuationRing_of_dedekind_domain S hq _
q.ramificationIdx R =
(IsDiscreteValuationRing.addVal (Localization.AtPrime q)
(algebraMap R (Localization.AtPrime q) t)).toNat := by
let : IsDiscreteValuationRing (Localization.AtPrime q) :=
IsLocalization.AtPrime.isDiscreteValuationRing_of_dedekind_domain S hq _
rw [Ideal.ramificationIdx_eq (Ideal.span {t}) q,
Ideal.map_span, Set.image_singleton, length_quotient_span_eq_addVal ht]
theorem finite_over_coefficient_field_of_finite_length
{F A M : Type*} [Field F] [CommRing A] [Algebra F A] [IsLocalRing A]
[AddCommGroup M] [Module A M] [Module F M] [IsScalarTower F A M]
(ε : A →+* F) (hε : ∀ c : F, ε (algebraMap F A c) = c)
(hM : IsFiniteLength A M) : Module.Finite F M := by
have hlength := PiExponentJets.W24.length_eq_coefficient_length ε hε hM
have hF : IsFiniteLength F M := Module.length_ne_top_iff.mp (by
rw [← hlength]
exact Module.length_ne_top_iff.mpr hM)
let : IsNoetherian F M := (isFiniteLength_iff_isNoetherian_isArtinian.mp hF).1
infer_instance
theorem principal_quotient_finite_over_coefficient_field
{F A : Type*} [Field F] [CommRing A] [IsDomain A] [IsDiscreteValuationRing A]
[Algebra F A] (ε : A →+* F) (hε : ∀ c : F, ε (algebraMap F A c) = c)
{a : A} (ha : a ≠ 0) : Module.Finite F (A ⧸ Ideal.span {a}) := by
exact finite_over_coefficient_field_of_finite_length ε hε
(Module.length_ne_top_iff.mp (principal_colength_ne_top ha))
theorem principal_quotient_finrank_eq_order
{F A : Type*} [Field F] [CommRing A] [IsDomain A] [IsDiscreteValuationRing A]
[Algebra F A] (ε : A →+* F) (hε : ∀ c : F, ε (algebraMap F A c) = c)
{a : A} (ha : a ≠ 0) :
(Module.finrank F (A ⧸ Ideal.span {a}) : ℕ∞) = IsDiscreteValuationRing.addVal A a := by
let := principal_quotient_finite_over_coefficient_field ε hε ha
rw [← PiExponentJets.W24.module_length_eq_finrank_of_augmentation ε hε]
exact length_quotient_span_eq_addVal ha
noncomputable def residueCoefficientEquiv
(F A : Type*) [Field F] [IsAlgClosed F] [CommRing A] [Algebra F A]
[IsLocalRing A] [Algebra.IsIntegral F (IsLocalRing.ResidueField A)] :
F ≃ₐ[F] IsLocalRing.ResidueField A :=
AlgEquiv.ofBijective (Algebra.ofId F (IsLocalRing.ResidueField A))
IsAlgClosed.algebraMap_bijective_of_isIntegral
noncomputable def residueAugmentation
(F A : Type*) [Field F] [IsAlgClosed F] [CommRing A] [Algebra F A]
[IsLocalRing A] [Algebra.IsIntegral F (IsLocalRing.ResidueField A)] : A →+* F :=
(residueCoefficientEquiv F A).symm.toRingHom.comp (IsLocalRing.residue A)
@[simp] theorem residueAugmentation_algebraMap
(F A : Type*) [Field F] [IsAlgClosed F] [CommRing A] [Algebra F A]
[IsLocalRing A] [Algebra.IsIntegral F (IsLocalRing.ResidueField A)] (c : F) :
residueAugmentation F A (algebraMap F A c) = c := by
change (residueCoefficientEquiv F A).symm
(algebraMap A (IsLocalRing.ResidueField A) (algebraMap F A c)) = c
rw [← IsScalarTower.algebraMap_apply F A (IsLocalRing.ResidueField A)]
exact (residueCoefficientEquiv F A).symm.commutes c
theorem principal_quotient_finrank_eq_order_of_algClosed
{F A : Type*} [Field F] [IsAlgClosed F] [CommRing A] [IsDomain A]
[IsDiscreteValuationRing A] [Algebra F A]
[Algebra.IsIntegral F (IsLocalRing.ResidueField A)] {a : A} (ha : a ≠ 0) :
(Module.finrank F (A ⧸ Ideal.span {a}) : ℕ∞) = IsDiscreteValuationRing.addVal A a :=
principal_quotient_finrank_eq_order (residueAugmentation F A)
(residueAugmentation_algebraMap F A) ha
end PiExponent.CurveLocalOrder
end OAI
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