Vector bundle

Summary

In algebraic geometry, a vector bundle of rank $n$ on a scheme $X$ is usually identified with a locally free sheaf of rank $n$. This is a sheaf of modules locally isomorphic to $\mathcal{O}_X^{\oplus n}$. Such a sheaf is the sheaf of sections of a geometric vector bundle. This is a scheme over $X$ whose fibres are affine spaces with a linear structure. Vector bundles of rank one are the invertible sheaves, and tangent and cotangent bundles of smooth varieties are the main examples of higher rank.