Definition 27.6.2. Let $S$ be a scheme. A vector bundle $\pi : V \to S$ over $S$ is an affine morphism of schemes such that $\pi _*\mathcal{O}_ V$ is endowed with the structure of a graded $\mathcal{O}_ S$-algebra $\pi _*\mathcal{O}_ V = \bigoplus \nolimits _{n \geq 0} \mathcal{E}_ n$ such that $\mathcal{E}_0 = \mathcal{O}_ S$ and such that the maps
\[ \text{Sym}^ n(\mathcal{E}_1) \longrightarrow \mathcal{E}_ n \]
are isomorphisms for all $n \geq 0$. A morphism of vector bundles over $S$ is a morphism $f : V \to V'$ such that the induced map
\[ f^* : \pi '_*\mathcal{O}_{V'} \longrightarrow \pi _*\mathcal{O}_ V \]
is compatible with the given gradings.
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