---
tag: P4DZM975BV
kind: concept
name: Vector bundle
msc: [14F06]
specializes: [KPKND0K7W6]
sources:
  - url: https://en.wikipedia.org/wiki/Coherent_sheaf
---

## 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.

## Explanations

### L1 human

A vector bundle attaches a vector space of fixed dimension to each point of a space, varying continuously.
Locally it is a product of the space with a vector space, but globally it may twist.
In algebraic geometry one usually works with the sheaf of its sections, which is locally a direct sum of copies of the sheaf of functions.
The tangent bundle of a smooth variety is an example: at each point, it gives the tangent space.