Given a non-archimedean local field $K$, let $\mathcal O_K$ be the associated valuation ring and $k$ its residue field. According to this MO answer, we have short exact sequence
$$0 \to H^2(\mathcal O_K,\Bbb G_m) \to H^2(K,\Bbb G_m) \to H^1(k,\Bbb Q/\Bbb Z) \to 0$$
I'm trying to see where this exact sequence comes from. The context is to compute the Brauer group of a local field, as is done in the linked answer.
Here are some thoughts: let $X=\mathrm{Spec}(\mathcal O_K)$ and let $i$ be the inclusion of the closed point and $j$ be the inclusion of the generic point. Then we are in the situation of section 5.5 from these notes by Brunault, so we get an exact sequence of étale sheaves (prop 5.29): $$0 \to j_!j^*\Bbb G_m \to \Bbb G_m \to i_*i^* \Bbb G_m \to 0$$
This induces a long exact sequence of étala cohomology group, but it I'm not sure how to compute it.