Definition 27.6.1. Let $S$ be a scheme. Let $\mathcal{E}$ be a quasi-coherent $\mathcal{O}_ S$-module^{1}. The *vector bundle associated to $\mathcal{E}$* is

## 27.6 Vector bundles

Let $S$ be a scheme. Let $\mathcal{E}$ be a quasi-coherent sheaf of $\mathcal{O}_ S$-modules. By Modules, Lemma 17.21.6 the symmetric algebra $\text{Sym}(\mathcal{E})$ of $\mathcal{E}$ over $\mathcal{O}_ S$ is a quasi-coherent sheaf of $\mathcal{O}_ S$-algebras. Hence it makes sense to apply the construction of the previous section to it.

The vector bundle associated to $\mathcal{E}$ comes with a bit of extra structure. Namely, we have a grading

which turns $\pi _*\mathcal{O}_{\mathbf{V}(\mathcal{E})}$ into a graded $\mathcal{O}_ S$-algebra. Conversely, we can recover $\mathcal{E}$ from the degree $1$ part of this. Thus we define an abstract vector bundle as follows.

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.

An example of a vector bundle over $S$ is affine $n$-space $\mathbf{A}^ n_ S$ over $S$, see Definition 27.5.1. This is true because $\mathcal{O}_ S[T_1, \ldots , T_ n] = \text{Sym}(\mathcal{O}_ S^{\oplus n})$.

Lemma 27.6.3. The category of vector bundles over a scheme $S$ is anti-equivalent to the category of quasi-coherent $\mathcal{O}_ S$-modules.

**Proof.**
Omitted. Hint: In one direction one uses the functor $\underline{\mathop{\mathrm{Spec}}}_ S(\text{Sym}^*_{\mathcal{O}_ S}(-))$ and in the other the functor $(\pi : V \to S) \leadsto (\pi _*\mathcal{O}_ V)_1$ where the subscript indicates we take the degree $1$ part.
$\square$

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