Let $X$ be a smooth geometrically integral variety over a number field $k$, we have an exact sequence of complexes of $\mathrm{Gal}(\bar{k}/k)$-modules $$0 \rightarrow [\bar{k}^* \rightarrow 0] \rightarrow [\bar{k}(X)^* \rightarrow \mathrm{Div}(\bar{X})] \rightarrow [\bar{k}(X)^*/\bar{k}^* \rightarrow \mathrm{Div}(\bar{X})] \rightarrow 0.$$

As usual, the complexes in the middle (called *1-motives*) are placed in degrees $-1$ and $0$. Taking hypercohomology groups, Lemma 2.1 of *Local-global principles for 1-motives* by Harari and Szamuely gives us the isomorphism $$\mathbf{H}^1(k,[\bar{k}(X)^* \rightarrow \mathrm{Div}(\bar{X})]) \cong \mathrm{Br}_1(X),$$
this is the *algebraic Brauer group* of $X$. I have two questions regarding its proof:

**(1)** It says that we have a distinguished triangle
$$\bar{k}(X)^* \rightarrow \mathrm{Div}(\bar{X}) \rightarrow [\bar{k}(X)^* \rightarrow \mathrm{Div}(\bar{X})] \rightarrow \bar{k}(X)^*[1],$$
but how do we check which exact triangle is this isomorphic to? This triangle itself is clearly not exact.

**(2)** The result seem to follow directly from the fact that the permutation module $\mathrm{Div}(\bar{X})$ has trivial $H^1$ (Galois cohomology), but I can't seem to see why.

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