in $A$. The sequence $$0 \to H^2_{\mathrm{\acute{e}t}}(A, \mu_\ell) \to H^2(K, \mu_\ell) \to \oplus_{P \in
  Spec(A)^{(1)}} H^1(\kappa(P), {\mathbb Z}/ \ell{\mathbb Z})$$ is exact, where Spec$(A)^{(1)}$ is the set of height one prime ideals of $A$.

We now recall a few notation from ([@hhk:patching_qf_csa]). Let $R$ be a complete discrete valuation ring with field of fractions $K$ and residue field $\kappa$. Let $F$ be the function field of a curve over $K$. Let ${{\mathscr{X}}}$ be a regular proper model of $F$ over $R$ and $X$ its reduced special fiber. For any codimension one point $\eta$ of ${{\mathscr{X}}}$, let $F_\eta$ be the completion of $F$ at the discrete valuation of $F$ given by $\eta$ and $\kappa(\eta)$ the residue field at $\eta$. For a closed point $P$ of ${{\mathscr{X}}}$, let $F_P$ be the field of fractions of the completion of the local ring at $P$ and $\kappa(P)$ the residue field at $P$. Let $U$ be an open subset of $X$. Let $R_U$ be the ring of all those functions in $F$ which are regular on $U$. Then $R
\subset R_U$. Let $t$ be a parameter in $R$. Let $\hat{R}_U$ be the completion of $R_U$ at the ideal $(t)$. Let $F_U$ be the field of 