theorem thm3_1_affine_iff_spec_g16 (k : Type v) [Field k] [IsAlgClosed k] : (∀ (X : AlgebraicGeometry.Scheme) (f : X ⟶ AlgebraicGeometry.Spec (CommRingCat.of k)) [Definition3_AlgebraicVariety k X f] [AlgebraicGeometry.IsAffine X], CategoryTheory.IsIso X.toSpecΓ ∧ IsReduced ↑(X.presheaf.obj (Opposite.op ⊤)) ∧ (CommRingCat.Hom.hom (AlgebraicGeometry.Scheme.Hom.appTop f)).FiniteType) ∧ ∀ (A : Type v) [inst : CommRing A] [inst_1 : Algebra k A] [Algebra.FiniteType k A] [IsReduced A], Definition3_AlgebraicVariety k (AlgebraicGeometry.Spec (CommRingCat.of A)) (AlgebraicGeometry.Spec.map (CommRingCat.ofHom (algebraMap k A)))

Lecture 3, Theorem 3.1: characterisation of affine algebraic varieties over an algebraically closed field as Spec A for reduced finitely generated k-algebras A.