File: Mathlib/AlgebraicGeometry/FunctionField.lean

full_name: AlgebraicGeometry.Scheme.functionField
code: /-- 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. -/
noncomputable abbrev Scheme.functionField [IrreducibleSpace X] : CommRingCat 
goal: CommRingCat

full_name: AlgebraicGeometry.Scheme.germToFunctionField
code: /-- The restriction map from a component to the function field. -/
noncomputable abbrev Scheme.germToFunctionField [IrreducibleSpace X] (U : X.Opens)
    [h : Nonempty U] : Γ(X, U) ⟶ X.functionField 
goal: X.presheaf.obj (Opposite.op U) ⟶ X.functionField

full_name: AlgebraicGeometry.instAlgebraαCommRingObjOppositeOpensTopologicalSpaceCarrierCommRingCatPresheafOpOpensFunctionFieldOfNonemptyToScheme
code: noncomputable instance [IrreducibleSpace X] (U : X.Opens) [Nonempty U] :
    Algebra Γ(X, U) X.functionField 
goal: Algebra ↑(X.presheaf.obj (Opposite.op U)) ↑X.functionField

full_name: AlgebraicGeometry.instFieldαCommRingFunctionField
code: noncomputable instance [IsIntegral X] : Field X.functionField 
goal: Field ↑X.functionField

full_name: AlgebraicGeometry.germ_injective_of_isIntegral
code: theorem germ_injective_of_isIntegral [IsIntegral X] {U : X.Opens} (x : U) :
    Function.Injective (X.presheaf.germ x) 
goal: Function.Injective ⇑(X.presheaf.germ x)

full_name: AlgebraicGeometry.Scheme.germToFunctionField_injective
code: theorem Scheme.germToFunctionField_injective [IsIntegral X] (U : X.Opens) [Nonempty U] :
    Function.Injective (X.germToFunctionField U) 
goal: Function.Injective ⇑(X.germToFunctionField U)

full_name: AlgebraicGeometry.genericPoint_eq_of_isOpenImmersion
code: theorem genericPoint_eq_of_isOpenImmersion {X Y : Scheme} (f : X ⟶ Y) [H : IsOpenImmersion f]
    [hX : IrreducibleSpace X] [IrreducibleSpace Y] :
    f.1.base (genericPoint X) = genericPoint Y 
goal: f.val.base (genericPoint ↑↑X.toPresheafedSpace) = genericPoint ↑↑Y.toPresheafedSpace

full_name: AlgebraicGeometry.stalkFunctionFieldAlgebra
code: noncomputable instance stalkFunctionFieldAlgebra [IrreducibleSpace X] (x : X) :
    Algebra (X.presheaf.stalk x) X.functionField 
goal: Algebra ↑(X.presheaf.stalk x) ↑X.functionField

full_name: AlgebraicGeometry.functionField_isScalarTower
code: instance functionField_isScalarTower [IrreducibleSpace X] (U : X.Opens) (x : U)
    [Nonempty U] : IsScalarTower Γ(X, U) (X.presheaf.stalk x) X.functionField 
goal: IsScalarTower ↑(X.presheaf.obj (Opposite.op U)) ↑(X.presheaf.stalk ↑x) ↑X.functionField

full_name: AlgebraicGeometry.instAlgebraαCommRingFunctionFieldSpec
code: noncomputable instance (R : CommRingCat.{u}) [IsDomain R] :
    Algebra R (Spec R).functionField 
goal: Algebra ↑R ↑(AlgebraicGeometry.Spec R).functionField

full_name: AlgebraicGeometry.genericPoint_eq_bot_of_affine
code: @[simp]
theorem genericPoint_eq_bot_of_affine (R : CommRingCat) [IsDomain R] :
    genericPoint (Spec R) = (⊥ : PrimeSpectrum R) 
goal: genericPoint ↑↑(AlgebraicGeometry.Spec R).toPresheafedSpace = ⊥

full_name: AlgebraicGeometry.functionField_isFractionRing_of_affine
code: instance functionField_isFractionRing_of_affine (R : CommRingCat.{u}) [IsDomain R] :
    IsFractionRing R (Spec R).functionField 
goal: IsFractionRing ↑R ↑(AlgebraicGeometry.Spec R).functionField

full_name: AlgebraicGeometry.instIsIntegralToSchemeOfNonemptyαTopologicalSpaceCarrierCommRingCat
code: instance {X : Scheme} [IsIntegral X] {U : X.Opens} [Nonempty U] :
    IsIntegral U 
goal: AlgebraicGeometry.IsIntegral ↑U

full_name: AlgebraicGeometry.IsAffineOpen.primeIdealOf_genericPoint
code: theorem IsAffineOpen.primeIdealOf_genericPoint {X : Scheme} [IsIntegral X] {U : X.Opens}
    (hU : IsAffineOpen U) [h : Nonempty U] :
    hU.primeIdealOf
        ⟨genericPoint X,
          ((genericPoint_spec X).mem_open_set_iff U.isOpen).mpr (by simpa using h)⟩ =
      genericPoint (Spec Γ(X, U)) 
goal: hU.primeIdealOf ⟨genericPoint ↑↑X.toPresheafedSpace, ⋯⟩ =
  genericPoint ↑↑(AlgebraicGeometry.Spec (X.presheaf.obj (Opposite.op U))).toPresheafedSpace

full_name: AlgebraicGeometry.functionField_isFractionRing_of_isAffineOpen
code: theorem functionField_isFractionRing_of_isAffineOpen [IsIntegral X] (U : X.Opens)
    (hU : IsAffineOpen U) [Nonempty U] :
    IsFractionRing Γ(X, U) X.functionField 
goal: IsFractionRing ↑(X.presheaf.obj (Opposite.op U)) ↑X.functionField

full_name: AlgebraicGeometry.instIsAffineObjAffineCover
code: instance (x : X) : IsAffine (X.affineCover.obj x) 
goal: AlgebraicGeometry.IsAffine (X.affineCover.obj x)

full_name: AlgebraicGeometry.instIsFractionRingαCommRingStalkCommRingCatPresheafFunctionField
code: instance [IsIntegral X] (x : X) :
    IsFractionRing (X.presheaf.stalk x) X.functionField 
goal: IsFractionRing ↑(X.presheaf.stalk x) ↑X.functionField