Gerbes and Brauer Groups - Core Definitions

Gerbes are stacks that are locally non-empty but have no global objects, generalizing torsors to the stack-theoretic setting. They classify twisted sheaves and central extensions.

DefinitionGerbe

A stack $\mathcal{G}$ over a site $\mathcal{C}$ is a gerbe if:

$\mathcal{G}$ is locally non-empty: there exists a covering $\{U_i \to *\}$ such that $\mathcal{G}(U_i) \neq \emptyset$ for each $i$
$\mathcal{G}$ is locally connected: for any $U$ and objects $x, y \in \mathcal{G}(U)$ , there exists a covering $\{V_j \to U\}$ such that $x|_{V_j} \simeq y|_{V_j}$ for all $j$

Equivalently, for any geometric point $\bar{x}$ , the fiber $\mathcal{G}_{\bar{x}}$ is a non-empty groupoid with all objects isomorphic (a connected groupoid).

ExampleThe Classifying Stack $\mathcal{B}G$

For an algebraic group $G$ , the stack $\mathcal{B}G$ is a gerbe over any base where it's defined. It has a single object (the trivial $G$ -torsor) at every geometric point, with automorphism group $G$ . This is the prototypical gerbe.

DefinitionBand of a Gerbe

The band (or lien) of a gerbe $\mathcal{G}$ is a sheaf of groups $\mathcal{A}$ such that for any $x \in \mathcal{G}(U)$ , we have: $\mathcal{A}|_U = \text{Aut}(x)$

The band is well-defined up to inner automorphism. When $\mathcal{A}$ is abelian, it's canonically defined. A gerbe is neutral if it has a global object, i.e., $\mathcal{G}(X) \neq \emptyset$ .

Example$\mathbb{G}_m$-Gerbes and Line Bundles

A gerbe bound by $\mathbb{G}_m$ (the multiplicative group) is equivalent to a class in $H^2(X, \mathbb{G}_m)$ . These classify:

Azumaya algebras of rank $n^2$
Central extensions of fundamental groups
Twisted sheaves and twisted K-theory

The neutral $\mathbb{G}_m$ -gerbes correspond to line bundles (via $H^1(X, \mathbb{G}_m) = \text{Pic}(X)$ ).

DefinitionBrauer Group

The Brauer group $\text{Br}(X)$ is defined as: $\text{Br}(X) = H^2_{\text{\'et}}(X, \mathbb{G}_m)$ in the étale topology. It classifies:

Azumaya algebras up to Morita equivalence
$\mathbb{G}_m$ -gerbes up to equivalence
Central simple algebras over $k(X)$ split by $X$

The Brauer group is a torsion abelian group fundamental in arithmetic geometry.

Remark

The Brauer group can also be defined cohomologically as: $0 \to \text{Pic}(X) \to H^2(X, \mathbb{G}_m) \to \text{Br}(X) \to 0$ This exact sequence shows $\text{Br}(X)$ measures obstructions to descending line bundles from coverings.

DefinitionTwisted Sheaves

Given a gerbe $\mathcal{G}$ bound by $\mathbb{G}_m$ , a $\mathcal{G}$ -twisted sheaf is a quasi-coherent sheaf on the gerbe $\mathcal{G}$ . Equivalently, it's a sheaf $\mathcal{F}$ on $X$ together with:

For each open $U$ and trivialization $\phi: \mathcal{G}|_U \simeq \mathcal{B}\mathbb{G}_m$ , an identification of $\mathcal{F}|_U$ with a $\mathbb{G}_m$ -module
Compatibility on overlaps governed by transition functions

Twisted sheaves form an abelian category $\textbf{Tw}(\mathcal{G})$ generalizing $\textbf{QCoh}(X)$ .

ExampleTwisted K-Theory

For a topological space $X$ with $H$ -flux $H \in H^3(X, \mathbb{Z})$ , the corresponding gerbe $\mathcal{G}_H$ has twisted K-theory: $K^*_{\mathcal{G}_H}(X) = K_0(\textbf{Tw}(\mathcal{G}_H))$

This appears in string theory and mathematical physics as the natural home for D-brane charges in the presence of $H$ -flux.

Gerbes provide the natural framework for understanding non-abelian cohomology, twisted structures, and obstructions in algebraic and arithmetic geometry.