Let $\mathcal{A}$ be an abelian sheaf of groups on a site $\mathcal{C}$. There is a bijection: $\{\text{equivalence classes of $\mathcal{A}$-gerbes on } X\} \leftrightarrow H^2(X, \mathcal{A})$

Moreover, automorphisms of a gerbe $\mathcal{G}$ are classified by $H^1(X, \mathcal{A})$, and the neutral gerbe $\mathcal{B}\mathcal{A}$ corresponds to $0 \in H^2(X, \mathcal{A})$.

This establishes gerbes as geometric realizations of second cohomology classes.

For a scheme $X$, there is an exact sequence: $0 \to \text{Pic}(X) \to H^2_{\text{Zar}}(X, \mathbb{G}_m) \to \text{Br}(X) \to H^1(X, \text{Pic})$

When $X$ is regular or quasi-projective, this simplifies to: $\text{Br}(X) = H^2_{\text{\'et}}(X, \mathbb{G}_m)_{\text{tors}}$

The Brauer group is always a torsion abelian group, and for proper varieties over algebraically closed fields, it's finite.

For any quasi-compact scheme $X$ with affine diagonal, the natural map: $\text{Br}'(X) \to \text{Br}(X)$ is an isomorphism, where $\text{Br}'(X)$ is the cohomological Brauer group. This resolves a long-standing question about different definitions of the Brauer group.

Let $X$ be a regular scheme and $Z \subset X$ a regular closed subscheme of codimension $d$. Then: $H^i_Z(X, \mathbb{G}_m) = \begin{cases} 0 & i < d \\ \mathbb{Z} & i = d \end{cases}$

For $d = 2$, this implies that elements of $\text{Br}(X)$ supported on smooth divisors are zero, providing strong constraints on Brauer classes.

There exist unirational threefolds that are not rational, with the obstruction lying in $\text{Br}(X)$. Specifically, they constructed a conic bundle $X \to \mathbb{P}^2$ with $\text{Br}(X)$ non-trivial, showing: $X(\mathbb{C}(t)) = \emptyset$ for the function field, proving non-rationality. This was the first non-rationality result using cohomological methods.

For a field $k$ of transcendence degree $d$ over an algebraically closed field: $\text{Br}(k) = 0 \text{ if } d \leq 1$

For function fields of curves over $\mathbb{C}$, the Brauer group vanishes. This has profound consequences for rationality questions and descent theory over function fields.

For a scheme $X$, there is an equivalence: $\{\text{Azumaya algebras on } X\}/\text{Morita} \simeq \text{Br}(X)$

Given $\alpha \in \text{Br}(X)$, there exists an Azumaya algebra $\mathcal{A}$ with $[\mathcal{A}] = \alpha$. The Morita equivalence class determines and is determined by the Brauer class, connecting non-commutative algebra to cohomology.

For a smooth geometrically rational surface $S$ over a number field $k$: $\text{Br}(S)/\text{Br}(k) \simeq (\mathbb{Q}/\mathbb{Z})^{\rho}$ where $\rho$ is the Picard rank. This explicit formula allows computation of Brauer groups for rational surfaces and has applications to the Brauer-Manin obstruction.