def IsVectorBundle {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) (n : ℕ) : Prop

Definition 28 (Lec 13): a morphism f : X → Y is a vector bundle of rank n if there is an open cover {Uᵢ} of Y such that, over each Uᵢ, f is isomorphic (over Uᵢ) to the trivial affine bundle 𝔸(Fin n; Uᵢ) → Uᵢ.

def IsLocallyFreeSheaf {X : AlgebraicGeometry.Scheme} (ℱ : X.Modules) (r : ℕ) : Prop

A sheaf of O_X-modules ℱ is locally free of rank r if X admits an open cover {Uᵢ} such that ℱ|_{Uᵢ} is isomorphic to the free O_{Uᵢ}-module of rank r. This is the sheaf counterpart of IsVectorBundle.