Definition
Fine Definition (the one I wrote down first): Let \(X\) be a scheme. We say that \(p:V\to X\) is a vector bundle or rank \(n\) over \(X\) if locally \(V\cong X\times \mathbb A^n\), if fiberwise \(V_x \cong \mathbb A^n\) has the structure of a vector space over \(k(x)\), and if the linear terms of \(p_*\mathcal O_V\) form a rank \(n\) locally free \(\mathcal O_X\)-module.
Better Definition (from Stacks Project): Let \(X\) be a scheme. A vector bundle over \(X\) is an affine morphism \(p:V\to X\) such that \(p_*\mathcal O_V\) has the structure of a graded \(\mathcal O_X\)-algebra \(p_*\mathcal O_X = \bigoplus_{r\geq 0} \mathcal E_n\) such that \(\mathcal E_0 = \mathcal O_X\) and that maps \(\Sym^r(\mathcal E_1)\to \mathcal E_r\) are isomorphisms for all \(r\).
One more definition that comes from equivalence of categories: Let \(X\) be a scheme and \(\mathcal F\) a locally free \(\mathcal O_X\)-module of rank \(n\). Then the vector bundle associated to \(\mathcal F\) is
\begin{align*} V = \Spec \Sym(\mathcal F). \end{align*}To dual or not to dual? Polynomial functions on a vector bundle \(V\) are \(\Sym (V^\vee)\); \(V^\vee\) is precisely the space of linear functions \(V\to \Spec k\), taking \(Sym(-)\) gives you the higher degree stuff. So think: \(\mathcal O_V = \Sym(V^\vee)\). Likewise, think “the degree 1 part of \(\mathcal O_V\) is \(V^\vee\)”.
Now consider sections of \(p:V\to X\). A section of \(p\) over an open set \(U\subseteq X\) is an \(X\)-morphism \(s:U\to V\), i.e. a \(\mathcal O_U\)-algebra homomorphism \(p_*\mathcal O_V|_U\to \mathcal O_U\). The relation is then
\begin{align*} \operatorname{Sect}(V/X) = \sHom_{\mathcal O_X}(p_*\mathcal O_V, \mathcal O_X) = (p_*\mathcal O_V)^\vee. \end{align*}Sections of \(p:V\to X\) are characters of \(p_*\mathcal O_V\).
So if you have a locally free \(\mathcal O_X\)-module \(\mathcal F\) and a vector bundle \(V\to X\) corresponding to it in some way, you have two options.
- If you want \(\mathcal F\) to be degree-one polynomial functions on \(V \), i.e. \(p_*\mathcal O_V\), then you want \(V = \Spec \Sym \mathcal F\).
- If you want \(\mathcal F\) to be the sheaf of sections of \(X\to V\), then you want \(V = \Spec Sym (\mathcal F^\vee)\).