def SectionsGenerate (X : AlgebraicGeometry.Scheme) (U V : X.Opens) : Prop

SectionsGenerate X U V asserts that on U ⊓ V the subring generated by restrictions of sections from U and from V is the whole ring (Lec 7, Prop 9 generation condition).

theorem isAffineHom_diagonal_iff_of_affine_target {X S : AlgebraicGeometry.Scheme} [AlgebraicGeometry.IsAffine S] {g : X ⟶ S} : AlgebraicGeometry.IsAffineHom (CategoryTheory.Limits.pullback.diagonal g) ↔ ∀ (V₁ V₂ : X.Opens), AlgebraicGeometry.IsAffineOpen V₁ → AlgebraicGeometry.IsAffineOpen V₂ → AlgebraicGeometry.IsAffineOpen (V₁ ⊓ V₂)

Over an affine base S, the diagonal Δ_g is affine iff intersections of affine opens in X remain affine.

theorem diagonal_surjective_iff_sectionsGenerate {X : AlgebraicGeometry.Scheme} {k : Type u} [Field k] (f : X ⟶ AlgebraicGeometry.Spec (CommRingCat.of k)) : (∀ (W : (CategoryTheory.Limits.pullback f f).Opens), AlgebraicGeometry.IsAffineOpen W → Function.Surjective ⇑(CategoryTheory.ConcreteCategory.hom (AlgebraicGeometry.Scheme.Hom.app (CategoryTheory.Limits.pullback.diagonal f) W))) ↔ ∀ (U V : X.Opens), AlgebraicGeometry.IsAffineOpen U → AlgebraicGeometry.IsAffineOpen V → SectionsGenerate X U V

Equivalence reformulating diagonal surjectivity on affine opens of the fiber square as a section-generation condition on intersections (Lec 7, Prop 9).

theorem prop9_separated_imp_affine_intersection {X : AlgebraicGeometry.Scheme} {k : Type u} [Field k] (f : X ⟶ AlgebraicGeometry.Spec (CommRingCat.of k)) [AlgebraicGeometry.IsSeparated f] {U V : X.Opens} (hU : AlgebraicGeometry.IsAffineOpen U) (hV : AlgebraicGeometry.IsAffineOpen V) : AlgebraicGeometry.IsAffineOpen (U ⊓ V)

Lec 7, Prop 9 (forward direction): if f : X → Spec k is separated then intersections of affine opens of X are affine.

theorem prop9_affine_diagonal_iff_affine_intersection {X : AlgebraicGeometry.Scheme} {k : Type u} [Field k] (f : X ⟶ AlgebraicGeometry.Spec (CommRingCat.of k)) : AlgebraicGeometry.IsAffineHom (CategoryTheory.Limits.pullback.diagonal f) ↔ ∀ (U V : X.Opens), AlgebraicGeometry.IsAffineOpen U → AlgebraicGeometry.IsAffineOpen V → AlgebraicGeometry.IsAffineOpen (U ⊓ V)

For f : X → Spec k, the diagonal is affine iff intersections of affine opens of X are affine (Lec 7, Prop 9).

theorem prop9_separated_iff_affine_intersection_and_tensor_surjective {X : AlgebraicGeometry.Scheme} {k : Type u} [Field k] (f : X ⟶ AlgebraicGeometry.Spec (CommRingCat.of k)) : AlgebraicGeometry.IsSeparated f ↔ (∀ (U V : X.Opens), AlgebraicGeometry.IsAffineOpen U → AlgebraicGeometry.IsAffineOpen V → AlgebraicGeometry.IsAffineOpen (U ⊓ V)) ∧ ∀ (U V : X.Opens), AlgebraicGeometry.IsAffineOpen U → AlgebraicGeometry.IsAffineOpen V → SectionsGenerate X U V

Lec 7, Prop 9: f : X → Spec k is separated iff (i) intersections of affine opens of X are affine and (ii) on every such intersection the restrictions from each side generate the ring of sections.