nLab
algebraic line n-bundle

Context

Bundles

Complex geometry

Contents

Idea

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

Properties

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

H n(X,𝔾 m)=π 0H(X,B n𝔾 m) H^n(X,\mathbb{G}_m) = \pi_0 \mathbf{H}(X,\mathbf{B}^n \mathbb{G}_m)

are all torsion groups for n2n \geq 2. (For n=2n = 2 this is the Brauer group.) See also this MO discussion.

See also at Friedlander-Milnor isomorphism conjecture.

References

Last revised on September 3, 2014 at 16:27:25. See the history of this page for a list of all contributions to it.