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Brauer group

Definition: \((\mathop{\mathrm{Br}}(k), \otimes_k)\) is the group of central simple algebras algebras over \(k\), up to Morita equivalence.

A higher invariant that extends the Picard group? Computed via descent techniques?

Measures complexity of a field by asking how many iso classes of finite dimensional central simple algebra exist over \(k\).

For fields, equivalent to Galois cohomology:

More generally, \(H^2(X;{\mathbf{G}}_m)\) for \(X\in {\mathsf{Sch}}\) classifies gerbes over \(X\) with structure group \({\mathbf{G}}_m\).

Finiteness of the Brauer group for scheme-theoretic surfaces is equivalent to the Tate conjecture for divisors on \(X\), i.e. finiteness of Sha of the Jacobian variety of the general fiber.

Used in the definition of the Brauer-Manin obstruction which obstructs lifting adelic points to Unsorted/rational points.

See spectral Brauer group