theorem lemma27_affine_morphism_iff {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) : AlgebraicGeometry.IsAffineHom f ↔ ∀ (U : Y.Opens), AlgebraicGeometry.IsAffineOpen U → AlgebraicGeometry.IsAffineOpen ((TopologicalSpace.Opens.map f.base).obj U)

Lemma 27 (Lec 13): a morphism of schemes f : X → Y is affine iff the preimage of every affine open of Y is an affine open of X.

theorem lemma27_affine_preimage {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [AlgebraicGeometry.IsAffineHom f] (U : Y.Opens) (hU : AlgebraicGeometry.IsAffineOpen U) : AlgebraicGeometry.IsAffineOpen ((TopologicalSpace.Opens.map f.base).obj U)

Forward direction of Lemma 27: if f is affine, then preimages of affine opens are affine opens.

theorem lemma27_affineHom_of_preimages_affine {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) (h : ∀ (U : Y.Opens), AlgebraicGeometry.IsAffineOpen U → AlgebraicGeometry.IsAffineOpen ((TopologicalSpace.Opens.map f.base).obj U)) : AlgebraicGeometry.IsAffineHom f

Converse direction of Lemma 27: if preimages of all affine opens are affine, then f is an affine morphism.

theorem lemma27_finite_morphism_iff {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) : AlgebraicGeometry.IsFinite f ↔ AlgebraicGeometry.IsAffineHom f ∧ ∀ (U : Y.Opens), AlgebraicGeometry.IsAffineOpen U → (CommRingCat.Hom.hom (AlgebraicGeometry.Scheme.Hom.app f U)).Finite

A morphism f : X → Y is finite iff it is affine and, for every affine open U of Y, the induced ring map Γ(U, O_Y) → Γ(f⁻¹U, O_X) is finite.

theorem lemma27_finite_is_affine {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [hf : AlgebraicGeometry.IsFinite f] : AlgebraicGeometry.IsAffineHom f

Every finite morphism is affine.

theorem lemma27_finite_ringHom_of_finite {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [hf : AlgebraicGeometry.IsFinite f] (U : Y.Opens) (hU : AlgebraicGeometry.IsAffineOpen U) : (CommRingCat.Hom.hom (AlgebraicGeometry.Scheme.Hom.app f U)).Finite

If f is finite then, on every affine open of Y, the corresponding map of section rings is a finite ring map.

theorem lemma27_finite_of_affine_and_finite_ringHom {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) (hAff : AlgebraicGeometry.IsAffineHom f) (hFin : ∀ (U : Y.Opens), AlgebraicGeometry.IsAffineOpen U → (CommRingCat.Hom.hom (AlgebraicGeometry.Scheme.Hom.app f U)).Finite) : AlgebraicGeometry.IsFinite f

Converse: an affine morphism whose induced ring maps on affine opens are all finite is itself a finite morphism.

theorem lemma27_finite_specMap_iff {R S : CommRingCat} (f : R ⟶ S) : AlgebraicGeometry.IsFinite (AlgebraicGeometry.Spec.map f) ↔ (CommRingCat.Hom.hom f).Finite

Affine case: Spec.map f : Spec S → Spec R is finite iff the underlying ring map f : R → S is finite.

theorem lemma27_finite_iff_integral_and_finiteType {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) : AlgebraicGeometry.IsFinite f ↔ AlgebraicGeometry.IsIntegralHom f ∧ AlgebraicGeometry.LocallyOfFiniteType f

A morphism is finite iff it is both integral and locally of finite type.