(see also Chern-Weil theory, parameterized homotopy theory)
vector bundle, (∞,1)-vector bundle
topological vector bundle, differentiable vector bundle, algebraic vector bundle
direct sum of vector bundles, tensor product of vector bundles, inner product of vector bundles?, dual vector bundle
geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
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
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
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.
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)
Last revised on September 3, 2014 at 16:27:25. See the history of this page for a list of all contributions to it.