class AlgebraicGeometry.Scheme.IsAlgebraicVariety (k : Type u) [Field k] (X : Scheme) [X.Over (Spec (CommRingCat.of k))] : Prop

An algebraic variety over k (Definition 3 of Lecture 2): a reduced scheme X over Spec k that is locally of finite type and admits a finite affine open cover.

• isReduced : IsReduced X
• locallyOfFiniteType : LocallyOfFiniteType (X ↘ Spec (CommRingCat.of k))
• hasFiniteAffineCover : ∃ (ι : Type u) (x : Fintype ι) (U : ι → X.Opens), (∀ (i : ι), IsAffineOpen (U i)) ∧ ⨆ (i : ι), U i = ⊤

instance AlgebraicGeometry.Scheme.IsAffineVariety.toIsAlgebraicVariety {k : Type u} [Field k] {X : Scheme} [X.Over (Spec (CommRingCat.of k))] [h : IsAffineVariety k X] : IsAlgebraicVariety k X

Every affine variety over k is in particular an algebraic variety, with the trivial finite affine cover by X itself.