# nLab algebraic line n-bundle

### Context

#### Bundles

bundles

fiber bundles in physics

complex geometry

# Contents

## Idea

The concept of line n-bundle in algebraic geometry, classified by maps into the $n$-fold delooping $\mathbf{B}^n \mathbb{G}_m$ of the multiplicative group.

## Properties

According to (Grothendieck 64, prop. 1.4) for $X$ a Noetherian scheme whose local rings have strict Henselisations that are factorial (…explain…) then the cohomology groups

$H^n(X,\mathbb{G}_m) = \pi_0 \mathbf{H}(X,\mathbf{B}^n \mathbb{G}_m)$

are all torsion groups for $n \geq 2$. (For $n = 2$ this is the Brauer group.) See also this MO discussion.