| 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 |
|
|
