nLab algebraic line n-bundle

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.

• for $n = 0$, classified by the group of units;

• for $n= 1$ these are algebraic line bundles, classified by Picard group, modulated by Picard stack;

• for $n =2$ these are algebraic line 2-bundles, classified by Brauer group, modulated by Brauer stack

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.

See also at Friedlander-Milnor isomorphism conjecture.

Related concepts

• holomorphic line n-bundle

References

• Alexander Grothendieck, Torsion homologique et sections rationnelles, Séminaire Claude Chevalley, 3 (1958), Exp. No. 5, 29 p. (Numdam)

• Alexander Grothendieck, Le groupe de Brauer : II. Théories cohomologiques. Séminaire Bourbaki, 9 (1964-1966), Exp. No. 297, 21 p. (Numdam)