openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /Analysis /AlgebraicLogarithmicObstruction.lean
| 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 | |