Vector bundle

Remarks

The free sheaf $\mathcal{O}_X^{\oplus n}$ is a vector bundle of rank $n$, with $U = X$ for every point. On $\mathbb{A}^1_k = \operatorname{Spec} k[t]$ over a field $k$, the skyscraper sheaf $\widetilde{M}$ with $M = k[t]/(t)$ is not a vector bundle. Its stalk at the origin is $k$, which is killed by $t$ and so is not a free module over the local ring $k[t]_{(t)}$; its stalks elsewhere are $0$. A vector bundle is a quasi-coherent sheaf, and it is coherent when $X$ is locally Noetherian. To $\mathcal{E}$ corresponds the geometric vector bundle $\mathbb{V}(\mathcal{E}^\vee) = \operatorname{Spec}_X \operatorname{Sym}(\mathcal{E}^\vee)$, whose sheaf of sections is $\mathcal{E}$. Hartshorne writes $\mathbb{V}(\mathcal{E}) = \operatorname{Spec}_X \operatorname{Sym}(\mathcal{E})$, whose sheaf of sections is $\mathcal{E}^\vee$, so the placement of the dual differs between sources.