algebraic line n-bundle in nLab

Contents

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.

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

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

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

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 n≥2n \geq 2. (For n=2n = 2 this is the Brauer group.) See also this MO discussion.

See also at Friedlander-Milnor isomorphism conjecture.

• 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)