Gerbes and Brauer Groups - Key Properties

Gerbes and Brauer groups exhibit rich structure connecting geometry, cohomology, and algebra. Understanding their properties is essential for applications.

TheoremClassification of Gerbes

Gerbes bound by an abelian sheaf of groups $\mathcal{A}$ on $X$ are classified by $H^2(X, \mathcal{A})$ . More precisely:

Equivalence classes of $\mathcal{A}$ -gerbes correspond to $H^2(X, \mathcal{A})$
Automorphisms of a gerbe $\mathcal{G}$ correspond to $H^1(X, \mathcal{A})$
The neutral gerbe $\mathcal{B}\mathcal{A}$ corresponds to $0 \in H^2(X, \mathcal{A})$

This establishes gerbes as geometric representatives of cohomology classes.

DefinitionCohomological Brauer Group

For a scheme $X$ , define: $\text{Br}'(X) = H^2_{\text{\'et}}(X, \mathbb{G}_m)_{\text{tors}}$ the torsion subgroup. When $X$ is regular, $\text{Br}'(X) = \text{Br}(X)$ . The Kummer sequence: $0 \to \mu_n \to \mathbb{G}_m \xrightarrow{n} \mathbb{G}_m \to 0$ gives: $\text{Pic}(X)/n \to H^2(X, \mu_n) \to {}_n\text{Br}(X) \to 0$ relating Brauer classes to roots of line bundles.

TheoremBrauer-Severi Varieties

A Brauer-Severi variety over a field $k$ is a variety $X$ such that $X_{\bar{k}} \simeq \mathbb{P}^{n-1}_{\bar{k}}$ . These are classified by $H^1(k, PGL_n)$ , which injects into $\text{Br}(k)$ via: $[X] \mapsto [\text{Azumaya algebra of endomorphisms}]$

Brauer-Severi varieties provide geometric representatives of Brauer classes.

DefinitionIndex and Period

For $\alpha \in \text{Br}(k)$ :

The period is the order of $\alpha$ in $\text{Br}(k)$
The index is the degree $[D : k]$ of the associated division algebra $D$

Index divides period, and both are equal over global fields (class field theory). Over function fields, index can be strictly smaller than period, providing subtle arithmetic invariants.

TheoremAzumaya Algebra Correspondence

There is a bijection: $\{\text{Azumaya algebras of rank } n^2\}/\text{Morita} \leftrightarrow \text{Br}(X)[n]$ The Morita equivalence classes of rank $n^2$ Azumaya algebras correspond to $n$ -torsion in the Brauer group. This connects non-commutative algebra to cohomological invariants.

DefinitionDerived Category of Twisted Sheaves

For a gerbe $\mathcal{G}$ on $X$ , the derived category $D^b(\mathcal{G})$ of twisted sheaves has properties:

It's a triangulated category with Serre duality
For $\alpha \in \text{Br}(X)$ , we have $D^b(\mathcal{G}_\alpha) \simeq D^b(\mathcal{G}_\beta)$ if $\alpha = \beta$ in $\text{Br}(X)$
Fourier-Mukai functors between twisted derived categories encode geometric correspondences

This framework is essential for homological mirror symmetry with Brauer classes.

TheoremMerkurjev-Suslin Theorem

For a field $k$ and prime $\ell$ : $K_2(k)/\ell \xrightarrow{\sim} {}_\ell\text{Br}(k)$ via the norm residue symbol. This connects K-theory to the Brauer group and is foundational in motivic cohomology. The full Bloch-Kato conjecture (now Voevodsky-Rost theorem) generalizes this to all $K_n(k)/\ell$ .

Remark

The Brauer group controls rational points: for $X$ over a global field $k$ , obstructions to the Hasse principle and weak approximation lie in $\text{Br}(X)$ . The Brauer-Manin obstruction: $X(k) \subset X(\mathbb{A}_k)^{\text{Br}}$ is a powerful tool in arithmetic geometry, explaining failures of local-to-global principles.

These properties show that gerbes and Brauer groups are fundamental invariants connecting geometry, cohomology, arithmetic, and derived categories.