@[reducible, inline]

abbrev VarietyMorphism (X Y : AlgebraicGeometry.Scheme) : Type u

A morphism between varieties, defined here as a scheme morphism X ⟶ Y.

theorem VarietyMorphism.continuous_base {X Y : AlgebraicGeometry.Scheme} (f : VarietyMorphism X Y) : Continuous ⇑f

The underlying map of a morphism of varieties is continuous.

def VarietyMorphism.pullbackSections {X Y : AlgebraicGeometry.Scheme} (f : VarietyMorphism X Y) (U : Y.Opens) : Y.presheaf.obj (Opposite.op U) ⟶ X.presheaf.obj (Opposite.op ((TopologicalSpace.Opens.map f.base).obj U))

Pullback of regular sections along a morphism: a function on U ⊆ Y pulls back to a function on the preimage f⁻¹(U) ⊆ X.

def VarietyMorphism.pullbackGlobal {X Y : AlgebraicGeometry.Scheme} (f : VarietyMorphism X Y) : Y.presheaf.obj (Opposite.op ⊤) ⟶ X.presheaf.obj (Opposite.op ⊤)

Pullback of global regular sections along a morphism of varieties.

theorem VarietyMorphism.pullback_id {X : AlgebraicGeometry.Scheme} (U : X.Opens) : pullbackSections (CategoryTheory.CategoryStruct.id X) U = CategoryTheory.CategoryStruct.id (X.presheaf.obj (Opposite.op U))

Pullback along the identity morphism is the identity.

theorem VarietyMorphism.pullback_comp {X Y Z : AlgebraicGeometry.Scheme} (f : VarietyMorphism X Y) (g : VarietyMorphism Y Z) (U : Z.Opens) : pullbackSections (CategoryTheory.CategoryStruct.comp f g) U = CategoryTheory.CategoryStruct.comp (g.pullbackSections U) (f.pullbackSections ((TopologicalSpace.Opens.map g.base).obj U))

Pullback of sections is contravariantly functorial: pullback along f ≫ g equals pullback along g followed by pullback along f.

def VarietyMorphism.toLocallyRingedSpaceHom {X Y : AlgebraicGeometry.Scheme} (f : VarietyMorphism X Y) : X.toLocallyRingedSpace ⟶ Y.toLocallyRingedSpace

The underlying morphism of locally ringed spaces of a morphism of varieties.