| import Mathlib |
| import OAI.AlgebraicGeometry.SectionFields.TensorAlgebra |
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| namespace OAI |
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| /-! Algebraic scalar extensions and finite Galois fields of definition. -/ |
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| noncomputable section |
| open AlgebraicGeometry CategoryTheory CategoryTheory.Limits TopologicalSpace Order Polynomial |
| open scoped TensorProduct WithZero |
| universe u |
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| namespace RelativeDenominators |
| open scoped TensorProduct |
| variable (K A L M : Type*) [Field K] [CommRing A] [Field L] [Field M] |
| [Algebra K A] [Algebra K L] [Algebra K M] [Algebra A M] [IsScalarTower K A M] |
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| @[instance_reducible] noncomputable def tensorAlgebraicFractionField |
| [IsDomain (A ⊗[K] L)] [IsFractionRing A M] [Algebra.IsAlgebraic K L] : |
| Field (M ⊗[K] L) := by |
| letI := tensorFraction_domain K A L M |
| letI : Algebra.IsIntegral K L := (Algebra.isAlgebraic_iff_isIntegral).mp inferInstance |
| exact (isField_of_isIntegral_of_isField' (R := M) (S := M ⊗[K] L) |
| (Field.toIsField M)).toField |
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| lemma tensorAlgebraicFraction_isFractionRing |
| [IsDomain (A ⊗[K] L)] [IsFractionRing A M] [Algebra.IsAlgebraic K L] : |
| letI := tensorFractionAlgebra K A L M |
| IsFractionRing (A ⊗[K] L) (M ⊗[K] L) := by |
| let := tensorFractionAlgebra K A L M |
| let := tensorFraction_isLocalization K A L M |
| let := tensorAlgebraicFractionField K A L M |
| have hsub : Algebra.algebraMapSubmonoid (A ⊗[K] L) (nonZeroDivisors A) ≤ |
| nonZeroDivisors (A ⊗[K] L) := by |
| rintro _ ⟨a, ha, rfl⟩ |
| rw [mem_nonZeroDivisors_iff_ne_zero] |
| intro h |
| have hn : Nontrivial A := (algebraMap A M).domain_nontrivial |
| apply nonZeroDivisors.ne_zero ha |
| apply Algebra.TensorProduct.includeLeft_injective (R := K) (S := K) (A := A) (B := L) |
| (RingHom.injective (algebraMap K L)) |
| change a ⊗ₜ[K] (1 : L) = (0 : A) ⊗ₜ[K] (1 : L) |
| change a ⊗ₜ[K] (1 : L) = 0 at h |
| simpa only [TensorProduct.zero_tmul] using h |
| have hinj : Function.Injective (algebraMap (A ⊗[K] L) (M ⊗[K] L)) := |
| IsLocalization.injective (M := Algebra.algebraMapSubmonoid (A ⊗[K] L) |
| (nonZeroDivisors A)) _ hsub |
| apply IsLocalization.of_le (Algebra.algebraMapSubmonoid (A ⊗[K] L) |
| (nonZeroDivisors A)) (nonZeroDivisors (A ⊗[K] L)) hsub |
| intro r hr |
| exact isUnit_iff_ne_zero.mpr (fun hz => nonZeroDivisors.ne_zero hr |
| (hinj (hz.trans (map_zero _).symm))) |
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| end RelativeDenominators |
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| open scoped TensorProduct |
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| namespace RelativeDenominators |
| section |
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| |
| def tensorIntermediateInclusion |
| (K A L : Type*) [Field K] [CommRing A] [Field L] [Algebra K A] [Algebra K L] |
| (E : IntermediateField K L) : A ⊗[K] E →ₐ[K] A ⊗[K] L := |
| Algebra.TensorProduct.map (AlgHom.id K A) E.val |
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| lemma tensorIntermediateInclusion_transition |
| (K A L : Type*) [Field K] [CommRing A] [Field L] [Algebra K A] [Algebra K L] |
| {E F : IntermediateField K L} (h : E ≤ F) (x : A ⊗[K] E) : |
| tensorIntermediateInclusion K A L F |
| (Algebra.TensorProduct.map (AlgHom.id K A) (IntermediateField.inclusion h) x) = |
| tensorIntermediateInclusion K A L E x := by |
| induction x using TensorProduct.inductionOn with |
| | tmul a e => rfl |
| | add x y hx hy => simp only [map_add, hx, hy] |
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| theorem tensor_finiteGalois_field_of_definition |
| (K A L : Type*) [Field K] [CommRing A] [Field L] [Algebra K A] [Algebra K L] |
| [IsGalois K L] (x : A ⊗[K] L) : |
| ∃ (E : FiniteGaloisIntermediateField K L) (y : A ⊗[K] E), |
| tensorIntermediateInclusion K A L E.toIntermediateField y = x := by |
| induction x using TensorProduct.inductionOn with |
| | tmul a l => |
| let E := FiniteGaloisIntermediateField.adjoin K ({l} : Set L) |
| have hl : l ∈ E.toIntermediateField := |
| FiniteGaloisIntermediateField.subset_adjoin K _ (Set.mem_singleton l) |
| exact ⟨E, a ⊗ₜ[K] (⟨l, hl⟩ : E), rfl⟩ |
| | add x y hx hy => |
| obtain ⟨E, x', hx'⟩ := hx |
| obtain ⟨F, y', hy'⟩ := hy |
| let G := E ⊔ F |
| have hE : E.toIntermediateField ≤ G.toIntermediateField := le_sup_left |
| have hF : F.toIntermediateField ≤ G.toIntermediateField := le_sup_right |
| refine ⟨G, Algebra.TensorProduct.map (AlgHom.id K A) |
| (IntermediateField.inclusion hE) x' + |
| Algebra.TensorProduct.map (AlgHom.id K A) |
| (IntermediateField.inclusion hF) y', ?_⟩ |
| rw [map_add, tensorIntermediateInclusion_transition, |
| tensorIntermediateInclusion_transition, hx', hy'] |
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| open scoped TensorProduct |
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| lemma tensorIntermediateInclusion_injective |
| (K A L : Type*) [Field K] [CommRing A] [Field L] [Algebra K A] [Algebra K L] |
| (E : IntermediateField K L) : |
| Function.Injective (tensorIntermediateInclusion K A L E) := by |
| change Function.Injective (LinearMap.lTensor A E.val.toLinearMap) |
| exact Module.Flat.lTensor_preserves_injective_linearMap E.val.toLinearMap E.val.injective |
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| lemma tensorIntermediate_domain |
| (K A L : Type*) [Field K] [CommRing A] [Field L] [Algebra K A] [Algebra K L] |
| [IsDomain (A ⊗[K] L)] (E : IntermediateField K L) : IsDomain (A ⊗[K] E) := |
| (tensorIntermediateInclusion_injective K A L E).isDomain |
| (tensorIntermediateInclusion K A L E).toRingHom |
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| def tensorIntermediateInclusionOver |
| (K A L M : Type*) [Field K] [CommRing A] [Field L] [CommRing M] |
| [Algebra K A] [Algebra K L] [Algebra K M] [Algebra A M] [IsScalarTower K A M] |
| (E : IntermediateField K L) : M ⊗[K] E →ₐ[A] M ⊗[K] L where |
| __ := (tensorIntermediateInclusion K M L E).toRingHom |
| commutes' a := by |
| change Algebra.TensorProduct.map (AlgHom.id K M) E.val |
| (algebraMap A M a ⊗ₜ[K] (1 : E)) = algebraMap A M a ⊗ₜ[K] (1 : L) |
| simp |
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| end |
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| open scoped TensorProduct |
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| |
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| theorem tensor_isIntegrallyClosed_of_algebraic_galois |
| (K A L M : Type*) [Field K] [CommRing A] [IsDomain A] [Field L] [Field M] |
| [Algebra K A] [Algebra K L] [Algebra K M] [Algebra A M] [IsScalarTower K A M] |
| [IsFractionRing A M] [IsIntegrallyClosed A] [IsGalois K L] |
| [IsDomain (A ⊗[K] L)] : IsIntegrallyClosed (A ⊗[K] L) := by |
| let := tensorFractionAlgebra K A L M |
| let := tensorFraction_isScalarTower K A L M |
| let := tensorAlgebraicFraction_isFractionRing K A L M |
| let := tensorAlgebraicFractionField K A L M |
| apply (isIntegrallyClosed_iff (M ⊗[K] L)).mpr |
| intro x hx |
| have hxA : IsIntegral A x := isIntegral_trans x hx |
| obtain ⟨E, y, hy⟩ := tensor_finiteGalois_field_of_definition K M L x |
| let := tensorIntermediate_domain K A L E.toIntermediateField |
| let := tensorFractionAlgebra K A E M |
| let := tensorFraction_isScalarTower K A E M |
| let := tensorFraction_isFractionRing K A E M |
| let := tensorFractionField K A E M |
| let := tensor_isIntegrallyClosed_of_galois K A E M |
| have hiy : IsIntegral A y := by |
| apply (isIntegral_algHom_iff (tensorIntermediateInclusionOver K A L M E.toIntermediateField) |
| (tensorIntermediateInclusion_injective K M L E.toIntermediateField)).mp |
| change IsIntegral A (tensorIntermediateInclusion K M L E.toIntermediateField y) |
| rw [hy] |
| exact hxA |
| obtain ⟨a, ha⟩ := (isIntegrallyClosed_iff (M ⊗[K] E)).mp |
| (inferInstance : IsIntegrallyClosed (A ⊗[K] E)) hiy.tower_top |
| refine ⟨tensorIntermediateInclusion K A L E.toIntermediateField a, ?_⟩ |
| rw [← hy, ← ha] |
| change Algebra.TensorProduct.map (IsScalarTower.toAlgHom K A M) (AlgHom.id K L) |
| (tensorIntermediateInclusion K A L E.toIntermediateField a) = |
| tensorIntermediateInclusion K M L E.toIntermediateField |
| (Algebra.TensorProduct.map (IsScalarTower.toAlgHom K A M) (AlgHom.id K E) a) |
| clear ha |
| induction a using TensorProduct.inductionOn with |
| | tmul a e => rfl |
| | add a b ha hb => simp only [map_add, ha, hb] |
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| end RelativeDenominators |
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| end |
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| end OAI |
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