nLab
Brauer stack

Context

Complex geometry

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

It is traditional to speak, for a suitable scheme XX, of its Picard group and of its Brauer group. Moreover, it is a classical fact that under suitable conditions the former admits itself a canonical geometric structure that makes it the Picard scheme of XX. Still well known, if maybe less commonly highlighted, is that this is just the 0-truncation of the Picard stack of XX, which is simply the mapping stack [X,B𝔾 m][X, \mathbf{B}\mathbb{G}_m] into the delooping of the multiplicative group. In this form this applies immediately also to more general context such as E-∞ geometry (“spectral geometry”) and gives a concept of Picard ∞-stack (“derived Picard stack”). Given this and the relation of the Brauer group to étale cohomology it is clear that the Brauer group similarly arises as the torsion subgroup of the 0-truncation of the ∞-stack which ought to be called the Brauer stack, given as the mapping stack

Br(X)[X,B 2𝔾 m] \mathbf{Br}(X) \coloneqq [X,\mathbf{B}^2 \mathbb{G}_m]

into the second delooping of the multiplicative group (modulating line 2-bundles, for instance holomorphic line 2-bundles in higher complex analytic geometry), where XX is a scheme or may itself be a derived scheme, algebraic stack, etc. Indeed, just as the Picard stack turns under Lie integration (evaluation on infinitesimally thickened points) and 0-truncation into what is commonly called the formal Picard group, so this Brauer \infty-stack similarly gives what is commonly called the formal Brauer group.

However, while therefore the terminology “Brauer stack” is the evident continuation of a traditional pattern (which in the other direction continues with the group of units and the mapping scheme [X,𝔾 m][X,\mathbb{G}_m]), it seems that this terminology has never been introduced in the literature (at time of this writing). (?)

References

Last revised on May 30, 2014 at 03:13:45. See the history of this page for a list of all contributions to it.