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

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

is compatible with the given gradings.

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