theorem Theorem2_2_isIso_toSpecΓ (k : Type u) [Field k] (X : AlgebraicGeometry.Scheme) (f : X ⟶ AlgebraicGeometry.Spec (CommRingCat.of k)) [Definition3_AlgebraicVariety k X f] [AlgebraicGeometry.IsAffine X] : CategoryTheory.IsIso X.toSpecΓ

Theorem 2.2 (first half): For an affine algebraic variety X over k, the canonical map X → Spec Γ(X, ⊤) is an isomorphism.

theorem Theorem2_2_globalSections_isReduced (k : Type u) [Field k] (X : AlgebraicGeometry.Scheme) (f : X ⟶ AlgebraicGeometry.Spec (CommRingCat.of k)) [hvar : Definition3_AlgebraicVariety k X f] [AlgebraicGeometry.IsAffine X] : IsReduced ↑(X.presheaf.obj (Opposite.op ⊤))

Theorem 2.2 (second half): For an affine algebraic variety, the global section ring Γ(X, ⊤) is reduced.

theorem Theorem2_2_globalSections_finiteType (k : Type u) [Field k] (X : AlgebraicGeometry.Scheme) (f : X ⟶ AlgebraicGeometry.Spec (CommRingCat.of k)) [hvar : Definition3_AlgebraicVariety k X f] [AlgebraicGeometry.IsAffine X] : (CommRingCat.Hom.hom (AlgebraicGeometry.Scheme.Hom.appTop f)).FiniteType

Theorem 2.2 (third half): For an affine algebraic variety, the global section ring Γ(X, ⊤) is a finitely generated k-algebra.

theorem Theorem2_2_affineVariety_iff_spec (k : Type u) [Field 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 u) [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)))

Theorem 2.2 (full equivalence): Affine algebraic varieties over k correspond exactly to spectra Spec A of finitely generated reduced k-algebras A.