@[reducible, inline]
Lecture 4, Definition 9: an affine morphism f : X → Y of schemes is one for which there
exists an affine open cover of Y whose preimages in X are affine.
Instances For
theorem
isAffineMorphism_iff
{X Y : AlgebraicGeometry.Scheme}
(f : X ⟶ Y)
:
IsAffineMorphism f ↔ ∀ (U : Y.Opens),
AlgebraicGeometry.IsAffineOpen U → AlgebraicGeometry.IsAffineOpen ((TopologicalSpace.Opens.map f.base).obj U)
A morphism is affine iff the preimage of every affine open of the target is affine in the source.
theorem
isAffineMorphism_of_affine_open_cover
{X Y : AlgebraicGeometry.Scheme}
(f : X ⟶ Y)
(H :
∀ (y : ↥Y),
∃ (U : Y.Opens),
y ∈ U ∧ AlgebraicGeometry.IsAffineOpen U ∧ AlgebraicGeometry.IsAffineOpen ((TopologicalSpace.Opens.map f.base).obj U))
:
It suffices to verify the affine-morphism condition pointwise: if every point of Y lies in
some affine open whose preimage is affine, then f is an affine morphism.