The Stacks project

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.

Definition 27.6.1. Let $S$ be a scheme. Let $\mathcal{E}$ be a quasi-coherent $\mathcal{O}_ S$-module1. The vector bundle associated to $\mathcal{E}$ is

\[ \mathbf{V}(\mathcal{E}) = \underline{\mathop{\mathrm{Spec}}}_ S(\text{Sym}(\mathcal{E})). \]

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

\[ \pi _*\mathcal{O}_{\mathbf{V}(\mathcal{E})} = \bigoplus \nolimits _{n \geq 0} \text{Sym}^ n(\mathcal{E}). \]

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

\[ \text{Sym}^ n(\mathcal{E}_1) \longrightarrow \mathcal{E}_ n \]

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.

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. $\square$

[1] The reader may expect here the condition that $\mathcal{E}$ is finite locally free. We do not do so in order to be consistent with [II, Definition 1.7.8, EGA].

Comments (4)

Comment #1787 by Arrow on

Shouldn't Lemma 26.6.3 state that geometric vector bundles over a fixed scheme are antiequivalent to locally free modules over it, instead of quasicoherent ones?

Comment #1789 by Matthieu Romagny on

Dear Arrow, you are right in some sense but did you read footnote 1 to Definition 26.6.1?

Comment #5317 by Alexander Duncan on

There is a bijection between the collection of all isomorphism classes of vector bundles of rank n over S to the collection of all isomorphism classes of locally free sheaves of rank n on S. This bijection is not particularly natural. The last exercise in section 2.5 of Hartshorne is giving this bijection.

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