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 : AlgebraicGeometry.Scheme) [IrreducibleSpace ↑↑X.toPresheafedSpace] : 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.toPresheafedSpace)

@[reducible, inline]

noncomputable abbrev AlgebraicGeometry.Scheme.germToFunctionField (X : AlgebraicGeometry.Scheme) [IrreducibleSpace ↑↑X.toPresheafedSpace] (U : TopologicalSpace.Opens ↑↑X.toPresheafedSpace) [h : Nonempty ↥U] : X.presheaf.obj { unop := U } ⟶ X.functionField

The restriction map from a component to the function field.

Equations

• X.germToFunctionField U = X.presheaf.germ ⟨genericPoint ↑↑X.toPresheafedSpace, ⋯⟩

noncomputable instance AlgebraicGeometry.instAlgebraαCommRingObjOppositeOpensTopologicalSpaceCarrierCommRingCatPresheafOpFunctionFieldOfNonemptySubtypeMem (X : AlgebraicGeometry.Scheme) [IrreducibleSpace ↑↑X.toPresheafedSpace] (U : TopologicalSpace.Opens ↑↑X.toPresheafedSpace) [Nonempty ↥U] : Algebra ↑(X.presheaf.obj { unop := U }) ↑X.functionField

Equations

• AlgebraicGeometry.instAlgebraαCommRingObjOppositeOpensTopologicalSpaceCarrierCommRingCatPresheafOpFunctionFieldOfNonemptySubtypeMem X U = RingHom.toAlgebra (X.germToFunctionField U)

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

Equations

• AlgebraicGeometry.instFieldαCommRingFunctionField X = Field.ofIsUnitOrEqZero ⋯

theorem AlgebraicGeometry.germ_injective_of_isIntegral (X : AlgebraicGeometry.Scheme) [AlgebraicGeometry.IsIntegral X] {U : TopologicalSpace.Opens ↑↑X.toPresheafedSpace} (x : ↥U) : Function.Injective ⇑(X.presheaf.germ x)

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

theorem AlgebraicGeometry.genericPoint_eq_of_isOpenImmersion {X : AlgebraicGeometry.Scheme} {Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [H : AlgebraicGeometry.IsOpenImmersion f] [hX : IrreducibleSpace ↑↑X.toPresheafedSpace] [IrreducibleSpace ↑↑Y.toPresheafedSpace] : f.val.base (genericPoint ↑↑X.toPresheafedSpace) = genericPoint ↑↑Y.toPresheafedSpace

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

Equations

• AlgebraicGeometry.stalkFunctionFieldAlgebra X x = RingHom.toAlgebra (X.presheaf.stalkSpecializes ⋯)

instance AlgebraicGeometry.functionField_isScalarTower (X : AlgebraicGeometry.Scheme) [IrreducibleSpace ↑↑X.toPresheafedSpace] (U : TopologicalSpace.Opens ↑↑X.toPresheafedSpace) (x : ↥U) [Nonempty ↥U] : IsScalarTower ↑(X.presheaf.obj { unop := U }) ↑(X.presheaf.stalk ↑x) ↑X.functionField

Equations

• ⋯ = ⋯

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

Equations

• One or more equations did not get rendered due to their size.

@[simp]

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

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

Equations

• ⋯ = ⋯

instance AlgebraicGeometry.instIsIntegralRestrictOfNonemptySubtypeαTopologicalSpaceCarrierCommRingCatMemOpens {X : AlgebraicGeometry.Scheme} [AlgebraicGeometry.IsIntegral X] {U : TopologicalSpace.Opens ↑↑X.toPresheafedSpace} [hU : Nonempty ↥U] : AlgebraicGeometry.IsIntegral (X ∣_ᵤ U)

Equations

• ⋯ = ⋯

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

theorem AlgebraicGeometry.functionField_isFractionRing_of_isAffineOpen (X : AlgebraicGeometry.Scheme) [AlgebraicGeometry.IsIntegral X] (U : TopologicalSpace.Opens ↑↑X.toPresheafedSpace) (hU : AlgebraicGeometry.IsAffineOpen U) [hU' : Nonempty ↥U] : IsFractionRing ↑(X.presheaf.obj { unop := U }) ↑X.functionField

instance AlgebraicGeometry.instIsAffineObjAffineCover (X : AlgebraicGeometry.Scheme) (x : ↑↑X.toPresheafedSpace) : AlgebraicGeometry.IsAffine (X.affineCover.obj x)

Equations

• ⋯ = ⋯

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

Equations

• ⋯ = ⋯