Witt residue sequence (${\partial }_K:\mathrm{Br}(K)\to H^1(k,\mathbb{Q}/\mathbb{Z})$)

For a complete discretely valued field $K$ with perfect residue field $k$, there is a split exact sequence $0\to \mathrm{Br}(k)\to \mathrm{Br}(K)\stackrel{{\partial }_K}{\to }{H}^1(k,\mathbb{Q}/\mathbb{Z})\to 0$. The residue is canonical. A uniformizer chooses a splitting $\chi \mapsto (\chi ,\pi )$; this choice does not make the residue itself noncanonical.

For a local field with finite residue field, evaluate the Witt residue character on arithmetic Frobenius. This gives ${\mathrm{inv}}_K:\mathrm{Br}(K)\cong \mathbb{Q}/\mathbb{Z}$. For the unramified degree-$m$ cyclic algebra with arithmetic Frobenius generator and parameter $a$, the invariant is $v_K(a)/m$. Its reduced denominator is the index of the division representative.

For a finite extension of local fields $L/K$, restriction multiplies the local Brauer invariant by $[L:K]$ and corestriction preserves it. For restriction, the residue-character value multiplies by the residue degree and the parameter valuation by the ramification index. For corestriction, lift the residue character using divisibility of $\mathbb{Q}/\mathbb{Z}$ and use the norm projection formula for cyclic Brauer pairings.