Function field of integral schemes

We define the function field of an irreducible scheme as the stalk of the generic point. This is a field when the scheme is integral.

Main definition

• AlgebraicGeometry.Scheme.functionField: The function field of an integral scheme.
• AlgebraicGeometry.Scheme.germToFunctionField: The canonical map from a component into the function field. This map is injective.

@[reducible, inline]

noncomputable abbrev AlgebraicGeometry.Scheme.functionField (X : Scheme) [IrreducibleSpace ↥X] : CommRingCat

The function field of an irreducible scheme is the local ring at its generic point. Despite the name, this is a field only when the scheme is integral.

Equations

• X.functionField = X.presheaf.stalk (genericPoint ↥X)

@[reducible, inline]

noncomputable abbrev AlgebraicGeometry.Scheme.germToFunctionField (X : Scheme) [IrreducibleSpace ↥X] (U : X.Opens) [h : Nonempty ↥↑U] : X.presheaf.obj (Opposite.op U) ⟶ X.functionField

The restriction map from a component to the function field.

Equations

• X.germToFunctionField U = X.presheaf.germ U (genericPoint ↥X) ⋯

noncomputable instance AlgebraicGeometry.instAlgebraCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensFunctionFieldOfNonemptyToScheme (X : Scheme) [IrreducibleSpace ↥X] (U : X.Opens) [Nonempty ↥↑U] : Algebra ↑(X.presheaf.obj (Opposite.op U)) ↑X.functionField

Equations

• AlgebraicGeometry.instAlgebraCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensFunctionFieldOfNonemptyToScheme X U = (CommRingCat.Hom.hom (X.germToFunctionField U)).toAlgebra

noncomputable instance AlgebraicGeometry.instFieldCarrierFunctionField (X : Scheme) [IsIntegral X] : Field ↑X.functionField

Equations

• AlgebraicGeometry.instFieldCarrierFunctionField X = Field.ofIsUnitOrEqZero ⋯

theorem AlgebraicGeometry.germ_injective_of_isIntegral (X : Scheme) [IsIntegral X] {U : X.Opens} (x : ↥X) (hx : x ∈ U) : Function.Injective ⇑(CategoryTheory.ConcreteCategory.hom (X.presheaf.germ U x hx))

theorem AlgebraicGeometry.Scheme.germToFunctionField_injective (X : Scheme) [IsIntegral X] (U : X.Opens) [Nonempty ↥↑U] : Function.Injective ⇑(CategoryTheory.ConcreteCategory.hom (X.germToFunctionField U))

theorem AlgebraicGeometry.genericPoint_eq_of_isOpenImmersion {X Y : Scheme} (f : X ⟶ Y) [H : IsOpenImmersion f] [hX : IrreducibleSpace ↥X] [IrreducibleSpace ↥Y] : (CategoryTheory.ConcreteCategory.hom f.base) (genericPoint ↥X) = genericPoint ↥Y

noncomputable instance AlgebraicGeometry.stalkFunctionFieldAlgebra (X : Scheme) [IrreducibleSpace ↥X] (x : ↥X) : Algebra ↑(X.presheaf.stalk x) ↑X.functionField

Equations

• AlgebraicGeometry.stalkFunctionFieldAlgebra X x = (CommRingCat.Hom.hom (X.presheaf.stalkSpecializes ⋯)).toAlgebra

instance AlgebraicGeometry.functionField_isScalarTower (X : Scheme) [IrreducibleSpace ↥X] (U : X.Opens) (x : ↥U) [Nonempty ↥↑U] : IsScalarTower ↑(X.presheaf.obj (Opposite.op U)) ↑(X.presheaf.stalk ↑x) ↑X.functionField

noncomputable instance AlgebraicGeometry.instAlgebraCarrierFunctionFieldSpec (R : CommRingCat) [IsDomain ↑R] : Algebra ↑R ↑(Spec R).functionField

Equations

• AlgebraicGeometry.instAlgebraCarrierFunctionFieldSpec R = (CommRingCat.Hom.hom (AlgebraicGeometry.StructureSheaf.toStalk (↑R) (genericPoint ↥(AlgebraicGeometry.Spec R)))).toAlgebra

@[simp]

theorem AlgebraicGeometry.genericPoint_eq_bot_of_affine (R : CommRingCat) [IsDomain ↑R] : genericPoint ↥(Spec R) = ⊥

instance AlgebraicGeometry.functionField_isFractionRing_of_affine (R : CommRingCat) [IsDomain ↑R] : IsFractionRing ↑R ↑(Spec R).functionField

instance AlgebraicGeometry.instIsIntegralToSchemeOfNonemptyCarrierCarrierCommRingCat {X : Scheme} [IsIntegral X] {U : X.Opens} [Nonempty ↥↑U] : IsIntegral ↑U

theorem AlgebraicGeometry.IsAffineOpen.primeIdealOf_genericPoint {X : Scheme} [IsIntegral X] {U : X.Opens} (hU : IsAffineOpen U) [h : Nonempty ↥↑U] : hU.primeIdealOf ⟨genericPoint ↥X, ⋯⟩ = genericPoint ↥(Spec (X.presheaf.obj (Opposite.op U)))

theorem AlgebraicGeometry.functionField_isFractionRing_of_isAffineOpen (X : Scheme) [IsIntegral X] (U : X.Opens) (hU : IsAffineOpen U) [Nonempty ↥↑U] : IsFractionRing ↑(X.presheaf.obj (Opposite.op U)) ↑X.functionField

instance AlgebraicGeometry.instIsAffineObjIsOpenImmersionAffineCover (X : Scheme) (x : ↥X) : IsAffine (X.affineCover.obj x)

instance AlgebraicGeometry.instIsFractionRingCarrierStalkCommRingCatPresheafFunctionField (X : Scheme) [IsIntegral X] (x : ↥X) : IsFractionRing ↑(X.presheaf.stalk x) ↑X.functionField