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import OAI.AlgebraicGeometry.SectionFields.TensorAlgebra
namespace OAI
/-! Algebraic scalar extensions and finite Galois fields of definition. -/
noncomputable section
open AlgebraicGeometry CategoryTheory CategoryTheory.Limits TopologicalSpace Order Polynomial
open scoped TensorProduct WithZero
universe u
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]
@[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
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)))
end RelativeDenominators
open scoped TensorProduct
namespace RelativeDenominators
section
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
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]
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']
open scoped TensorProduct
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
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
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
end
open scoped TensorProduct
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]
end RelativeDenominators
end
end OAI
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