File size: 5,027 Bytes
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 | import Mathlib
import OAI.NumberTheory.PiExponent.Geometry.CurveLocalOrder
namespace OAI
noncomputable section
open scoped nonZeroDivisors
open PiExponent.WeightedCurveDegree PiExponent.CurveLocalOrder
namespace PiExponent.DVRValuationUnique
variable {A E : Type*} [CommRing A] [IsDomain A] [IsDiscreteValuationRing A]
[Field E] [Algebra A E] [IsFractionRing A E]
def localFieldUnit (a : A) (ha : a ≠ 0) : Eˣ :=
Units.mk0 (algebraMap A E a) ((map_ne_zero_iff _ (IsFractionRing.injective A E)).mpr ha)
omit [IsDomain A] [IsDiscreteValuationRing A] in
@[simp] theorem localFieldUnit_val (a : A) (ha : a ≠ 0) :
((localFieldUnit (E := E) a ha : Eˣ) : E) = algebraMap A E a := rfl
omit [IsDomain A] [IsDiscreteValuationRing A] [IsFractionRing A E] in
theorem integerOrder_unit_eq_zero
(v : AddValuation E (WithTop ℤ))
(hregular : ∀ a : A, 0 ≤ v (algebraMap A E a)) (u : Aˣ) :
integerOrder v (Units.map (algebraMap A E) u) = 0 := by
have hpos : 0 ≤ integerOrder v (Units.map (algebraMap A E) u) := by
apply WithTop.coe_le_coe.mp
simpa using hregular (u : A)
have hneg : 0 ≤ integerOrder v (Units.map (algebraMap A E) u⁻¹) := by
apply WithTop.coe_le_coe.mp
simpa using hregular (u⁻¹ : Aˣ)
simp only [map_inv, integerOrder_inv] at hneg
omega
theorem exists_order_eq_uniformizer_multiple
(v : AddValuation E (WithTop ℤ))
(hregular : ∀ a : A, 0 ≤ v (algebraMap A E a))
{π : A} (hπ : Irreducible π) (a : A) (ha : a ≠ 0) :
∃ n : ℕ, integerOrder v (localFieldUnit (E := E) a ha) =
(n : ℤ) * integerOrder v (localFieldUnit (E := E) π hπ.ne_zero) ∧
IsDiscreteValuationRing.addVal A a = n := by
obtain ⟨n, u, rfl⟩ := IsDiscreteValuationRing.eq_unit_mul_pow_irreducible ha hπ
refine ⟨n, ?_, IsDiscreteValuationRing.addVal_def' u hπ n⟩
have he : localFieldUnit (E := E) ((u : A) * π ^ n) ha =
Units.map (algebraMap A E) u * localFieldUnit (E := E) π hπ.ne_zero ^ n := by
ext
simp [localFieldUnit, map_mul, map_pow]
rw [he, integerOrder_mul, integerOrder_pow, integerOrder_unit_eq_zero v hregular]
simp
theorem uniformizer_order_dvd
(v : AddValuation E (WithTop ℤ))
(hregular : ∀ a : A, 0 ≤ v (algebraMap A E a))
{π : A} (hπ : Irreducible π) (z : Eˣ) :
integerOrder v (localFieldUnit (E := E) π hπ.ne_zero) ∣ integerOrder v z := by
obtain ⟨a, b, hb, hab⟩ := IsFractionRing.div_surjective A (z : E)
have hb0 : b ≠ 0 := mem_nonZeroDivisors_iff_ne_zero.mp hb
have ha0 : a ≠ 0 := by
intro ha
apply z.ne_zero
rw [← hab, ha, map_zero, zero_div]
have hz : z = localFieldUnit (E := E) a ha0 / localFieldUnit (E := E) b hb0 := by
ext
simpa only [Units.val_div_eq_div_val, localFieldUnit_val] using hab.symm
obtain ⟨m, hm, _⟩ := exists_order_eq_uniformizer_multiple v hregular hπ a ha0
obtain ⟨n, hn, _⟩ := exists_order_eq_uniformizer_multiple v hregular hπ b hb0
rw [hz, integerOrder_div, hm, hn]
exact ⟨(m : ℤ) - n, by ring⟩
theorem eq_fractionAddValuation
(v : AddValuation E (WithTop ℤ))
(hregular : ∀ a : A, 0 ≤ v (algebraMap A E a))
(hcenter : ∀ a ∈ IsLocalRing.maximalIdeal A, a ≠ 0 → 0 < v (algebraMap A E a))
(hnormalized : ∃ z : Eˣ, integerOrder v z = 1) :
v = fractionAddValuation A E := by
obtain ⟨π, hπ⟩ := IsDiscreteValuationRing.exists_irreducible A
have hπmem : π ∈ IsLocalRing.maximalIdeal A := by
rw [hπ.maximalIdeal_eq]
exact Ideal.subset_span (Set.mem_singleton π)
have hpos : 0 < integerOrder v (localFieldUnit (E := E) π hπ.ne_zero) := by
apply WithTop.coe_lt_coe.mp
simpa using hcenter π hπmem hπ.ne_zero
have hone : integerOrder v (localFieldUnit (E := E) π hπ.ne_zero) = 1 := by
obtain ⟨z, hz⟩ := hnormalized
apply Int.eq_one_of_dvd_one hpos.le
rw [← hz]
exact uniformizer_order_dvd v hregular hπ z
have hlocal (a : A) : v (algebraMap A E a) = localAddValuation A a := by
by_cases ha : a = 0
· simp [ha]
obtain ⟨n, hn, hcanonical⟩ := exists_order_eq_uniformizer_multiple v hregular hπ a ha
rw [hone, mul_one] at hn
rw [localAddValuation_apply, hcanonical, enatToIntegerOrder_natCast]
rw [← localFieldUnit_val a ha, ← coe_integerOrder, hn]
rfl
ext x
obtain ⟨a, b, _, rfl⟩ := IsFractionRing.div_surjective A x
rw [AddValuation.map_div, AddValuation.map_div, hlocal, hlocal,
fractionAddValuation_algebraMap, fractionAddValuation_algebraMap]
theorem fractionAddValuation_normalized :
∃ z : Eˣ, integerOrder (fractionAddValuation A E) z = 1 := by
obtain ⟨π, hπ⟩ := IsDiscreteValuationRing.exists_irreducible A
refine ⟨localFieldUnit (E := E) π hπ.ne_zero, ?_⟩
apply WithTop.coe_injective
rw [coe_integerOrder, localFieldUnit_val, fractionAddValuation_algebraMap,
localAddValuation_apply, IsDiscreteValuationRing.addVal_uniformizer hπ]
rfl
end PiExponent.DVRValuationUnique
end
end OAI
|