| import OAI.NumberTheory.PiExponent.Geometry.CurveLocalOrder |
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| namespace OAI |
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| noncomputable section |
| namespace PiExponent.DVRBranchOrder |
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| variable {A C : Type*} [CommRing A] [IsDomain A] [IsDiscreteValuationRing A] [Field C] |
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| theorem order_eq_addVal (φ : A →+* PowerSeries C) (π : A) |
| (hπ : Irreducible π) (hφπ : φ π = PowerSeries.X) (a : A) : |
| PowerSeries.order (φ a) = IsDiscreteValuationRing.addVal A a := by |
| by_cases ha : a = 0 |
| · subst a |
| rw [map_zero, PowerSeries.order_zero, IsDiscreteValuationRing.addVal_zero] |
| · obtain ⟨n, u, rfl⟩ := IsDiscreteValuationRing.eq_unit_mul_pow_irreducible ha hπ |
| rw [map_mul, map_pow, hφπ, PowerSeries.order_mul, PowerSeries.order_X_pow, |
| PowerSeries.order_zero_of_isUnit (u.isUnit.map φ), zero_add, |
| IsDiscreteValuationRing.addVal_def' u hπ] |
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| theorem injective (φ : A →+* PowerSeries C) (π : A) |
| (hπ : Irreducible π) (hφπ : φ π = PowerSeries.X) : Function.Injective φ := by |
| intro a b hab |
| have hz : φ (a - b) = 0 := by rw [map_sub, hab, sub_self] |
| have ho := order_eq_addVal φ π hπ hφπ (a - b) |
| rw [hz, PowerSeries.order_zero] at ho |
| exact sub_eq_zero.mp (IsDiscreteValuationRing.addVal_eq_top_iff.mp ho.symm) |
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| theorem order_eq_top_iff (φ : A →+* PowerSeries C) (π : A) |
| (hπ : Irreducible π) (hφπ : φ π = PowerSeries.X) (a : A) : |
| PowerSeries.order (φ a) = ⊤ ↔ a = 0 := by |
| rw [order_eq_addVal φ π hπ hφπ, IsDiscreteValuationRing.addVal_eq_top_iff] |
| |
| theorem order_eq_zero_iff_isUnit (φ : A →+* PowerSeries C) (π : A) |
| (hπ : Irreducible π) (hφπ : φ π = PowerSeries.X) (a : A) : |
| PowerSeries.order (φ a) = 0 ↔ IsUnit a := by |
| rw [order_eq_addVal φ π hπ hφπ, IsDiscreteValuationRing.addVal_eq_zero_iff] |
| |
| theorem order_le_iff_dvd (φ : A →+* PowerSeries C) (π : A) |
| (hπ : Irreducible π) (hφπ : φ π = PowerSeries.X) (a b : A) : |
| PowerSeries.order (φ a) ≤ PowerSeries.order (φ b) ↔ a ∣ b := by |
| rw [order_eq_addVal φ π hπ hφπ, order_eq_addVal φ π hπ hφπ, |
| IsDiscreteValuationRing.addVal_le_iff_dvd] |
| |
| theorem order_eq_principal_colength (φ : A →+* PowerSeries C) (π : A) |
| (hπ : Irreducible π) (hφπ : φ π = PowerSeries.X) {a : A} (ha : a ≠ 0) : |
| PowerSeries.order (φ a) = Module.length A (A ⧸ Ideal.span {a}) := by |
| rw [order_eq_addVal φ π hπ hφπ, CurveLocalOrder.length_quotient_span_eq_addVal ha] |
|
|
| theorem fraction_order_eq_branch_order {E : Type*} [Field E] [Algebra A E] [IsFractionRing A E] |
| (φ : A →+* PowerSeries C) (π : A) (hπ : Irreducible π) |
| (hφπ : φ π = PowerSeries.X) (a : A) : |
| CurveLocalOrder.fractionAddValuation A E (algebraMap A E a) = |
| CurveLocalOrder.enatToIntegerOrder (PowerSeries.order (φ a)) := by |
| rw [CurveLocalOrder.fractionAddValuation_algebraMap, CurveLocalOrder.localAddValuation_apply, |
| order_eq_addVal φ π hπ hφπ] |
|
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| theorem integerOrder_eq_branch_order {E : Type*} [Field E] [Algebra A E] [IsFractionRing A E] |
| (φ : A →+* PowerSeries C) (π : A) (hπ : Irreducible π) |
| (hφπ : φ π = PowerSeries.X) {a : A} (ha : a ≠ 0) : |
| WeightedCurveDegree.integerOrder (CurveLocalOrder.fractionAddValuation A E) |
| (Units.mk0 (algebraMap A E a) ((map_ne_zero_iff _ (IsFractionRing.injective A E)).mpr ha)) = |
| ((PowerSeries.order (φ a)).toNat : ℤ) := by |
| rw [CurveLocalOrder.integerOrder_field_image_eq_length ha, |
| ← order_eq_principal_colength φ π hπ hφπ ha] |
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| variable [Algebra C A] |
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| theorem algHom_order_eq_addVal (φ : A →ₐ[C] PowerSeries C) (π : A) |
| (hπ : Irreducible π) (hφπ : φ π = PowerSeries.X) (a : A) : |
| PowerSeries.order (φ a) = IsDiscreteValuationRing.addVal A a := |
| order_eq_addVal φ.toRingHom π hπ hφπ a |
|
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| theorem algHom_injective (φ : A →ₐ[C] PowerSeries C) (π : A) |
| (hπ : Irreducible π) (hφπ : φ π = PowerSeries.X) : Function.Injective φ := |
| injective φ.toRingHom π hπ hφπ |
|
|
| end PiExponent.DVRBranchOrder |
| end |
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| end OAI |
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