File size: 6,025 Bytes
a72760f | 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 | import Mathlib.FieldTheory.AlgebraicClosure
import Mathlib.RingTheory.Etale.Field
import Mathlib.RingTheory.Etale.Kaehler
import OAI.NumberTheory.PiExponent.Analysis.LogarithmicObstruction
import OAI.NumberTheory.PiExponent.LocalAlgebra.TranscendenceBasisReduction
namespace OAI
noncomputable section
namespace PiExponent
open KaehlerDifferential
open scoped TensorProduct Polynomial nonZeroDivisors IntermediateField
def extendDerivation
{R S T : Type*} [CommRing R] [CommRing S] [CommRing T]
[Algebra R S] [Algebra R T] [Algebra S T] [IsScalarTower R S T]
[Algebra.FormallyEtale S T]
(d : Derivation R S S) : Derivation R T T :=
((LinearMap.liftBaseChange T ((Algebra.linearMap S T).comp d.liftKaehlerDifferential)).comp
(tensorKaehlerEquivOfFormallyEtale R S T).symm.toLinearMap).compDer (D R T)
theorem extendDerivation_algebraMap
{R S T : Type*} [CommRing R] [CommRing S] [CommRing T]
[Algebra R S] [Algebra R T] [Algebra S T] [IsScalarTower R S T]
[Algebra.FormallyEtale S T]
(d : Derivation R S S) (s : S) :
extendDerivation (T := T) d (algebraMap S T s) = algebraMap S T (d s) := by
simp [extendDerivation, LinearMap.compDer,
tensorKaehlerEquivOfFormallyEtale_symm_D_algebraMap,
Derivation.liftKaehlerDifferential_comp_D]
def ratFuncDerivative {K : Type*} [Field K] : Derivation K (RatFunc K) (RatFunc K) := by
letI : Algebra.FormallyEtale K[X] (RatFunc K) :=
Algebra.FormallyEtale.of_isLocalization (K[X]⁰)
exact extendDerivation Polynomial.derivative'
@[simp]
theorem ratFuncDerivative_X {K : Type*} [Field K] :
ratFuncDerivative (RatFunc.X : RatFunc K) = 1 := by
let : Algebra.FormallyEtale K[X] (RatFunc K) :=
Algebra.FormallyEtale.of_isLocalization (K[X]⁰)
change extendDerivation Polynomial.derivative' (RatFunc.X : RatFunc K) = 1
rw [← RatFunc.algebraMap_X, extendDerivation_algebraMap]
simp [Polynomial.derivative']
theorem finite_extension_no_logarithmic_differential
{K E : Type*} [Field K] [CharZero K] [Field E]
[Algebra K E] [Algebra (RatFunc K) E] [IsScalarTower K (RatFunc K) E]
[FiniteDimensional (RatFunc K) E] (x : E) :
D K E x ≠ (algebraMap (RatFunc K) E RatFunc.X)⁻¹ •
D K E (algebraMap (RatFunc K) E RatFunc.X) := by
let : Algebra.FormallyEtale (RatFunc K) E :=
Algebra.FormallyEtale.of_isSeparable (RatFunc K) E
let delta : Derivation K E E := extendDerivation ratFuncDerivative
have hcompat (a : RatFunc K) :
delta (algebraMap (RatFunc K) E a) =
algebraMap (RatFunc K) E (ratFuncDerivative a) :=
extendDerivation_algebraMap ratFuncDerivative a
intro heq
have hd := congrArg delta.liftKaehlerDifferential heq
simp only [Derivation.liftKaehlerDifferential_comp_D, map_smul, hcompat,
ratFuncDerivative_X, map_one, smul_eq_mul, mul_one] at hd
apply finite_extension_derivation_ne_simple_pole ratFuncDerivative delta
ratFuncDerivative_X hcompat x (c := (1 : K)) one_ne_zero
simpa only [map_div₀, map_one, one_div, map_inv₀] using hd
theorem adjoin_finite_no_logarithmic_differential
{K E : Type*} [Field K] [CharZero K] [Field E] [Algebra K E]
(y : E) (hy : Transcendental K y) [FiniteDimensional K⟮y⟯ E] (x : E) :
D K E x ≠ y⁻¹ • D K E y := by
let F := K⟮y⟯
let e : RatFunc K ≃ₐ[K] F := RatFunc.algEquivOfTranscendental y hy
let : Algebra (RatFunc K) F := e.toRingHom.toAlgebra
let : Algebra (RatFunc K) E :=
((algebraMap F E).comp e.toRingHom).toAlgebra
let : IsScalarTower (RatFunc K) F E := IsScalarTower.of_algebraMap_eq' rfl
let : IsScalarTower K (RatFunc K) E := IsScalarTower.of_algebraMap_eq fun a => by
change algebraMap K E a = algebraMap F E (e (algebraMap K (RatFunc K) a))
rw [e.commutes]
exact IsScalarTower.algebraMap_apply K F E a
have : Module.Finite (RatFunc K) F :=
Module.Finite.of_surjective (Algebra.linearMap (RatFunc K) F) e.surjective
have : Module.Finite (RatFunc K) E := Module.Finite.trans F E
have hX : algebraMap (RatFunc K) E RatFunc.X = y := by
change ((RatFunc.algEquivOfTranscendental y hy) RatFunc.X : E) = y
exact RatFunc.algEquivOfTranscendental_X y hy
simpa only [hX] using finite_extension_no_logarithmic_differential (K := K) x
theorem logarithmic_differential_base_change
{C K E : Type*} [Field C] [Field K] [Field E]
[Algebra C K] [Algebra C E] [Algebra K E] [IsScalarTower C K E]
{x y : E} (h : D C E x = y⁻¹ • D C E y) :
D K E x = y⁻¹ • D K E y := by
have hm := congrArg (KaehlerDifferential.map C K E E) h
simpa only [map_D, map_smul, Algebra.algebraMap_self, RingHom.id_apply] using hm
theorem no_logarithmic_differential_over_intermediateField
{C E : Type*} [Field C] [CharZero C] [Field E] [Algebra C E]
(K : IntermediateField C E) (y : E) (hy : Transcendental K y)
[FiniteDimensional K⟮y⟯ E] (x : E) :
D C E x ≠ y⁻¹ • D C E y := by
intro h
exact adjoin_finite_no_logarithmic_differential y hy x
(logarithmic_differential_base_change (K := K) h)
theorem isAlgebraic_of_logarithmic_differential
{C E : Type*} [Field C] [CharZero C] [Field E] [Algebra C E]
[Algebra.EssFiniteType C E] (x y : E)
(h : D C E x = y⁻¹ • D C E y) : IsAlgebraic C y := by
by_contra hy
obtain ⟨K, hKy, hfinite⟩ := exists_intermediateField_finite_over_simple C E hy
have : FiniteDimensional K⟮y⟯ E := hfinite
exact no_logarithmic_differential_over_intermediateField K y hKy x h
theorem constant_of_logarithmic_differential
{C E : Type*} [Field C] [CharZero C] [IsAlgClosed C] [Field E] [Algebra C E]
[Algebra.EssFiniteType C E] (x y : E)
(h : D C E x = y⁻¹ • D C E y) : ∃ c : C, algebraMap C E c = y := by
have hy := isAlgebraic_of_logarithmic_differential x y h
have hmem : y ∈ algebraicClosure C E := mem_algebraicClosure_iff.mpr hy
rw [IntermediateField.eq_bot_of_isAlgClosed_of_isAlgebraic (algebraicClosure C E)] at hmem
exact IntermediateField.mem_bot.mp hmem
end PiExponent
end
end OAI
|