$$0 \lrar {\mathbb{G}}_{m,X'} \lrar \nu'_*{\mathbb{G}}_{m,\wtl{X}} \oplus i'_*{\mathbb{G}}_{m,X_{\operatorname{sing}}} \lrar \nu'_*j_*{\mathbb{G}}_{m,\wtl{X}_{\operatorname{crit}}} \lrar 0$$ is exact, as follows. From the definition of $X'$ we obtain that the square $$\xymatrix{
\OO_{X'} \ar[r]\ar[d] & \nu'_*\OO_{\wtl{X}} \ar[d] \\
i'_*\OO_{X_{\operatorname{sing}}} \ar[r] & \nu'_*j_*\OO_{\wtl{X}_{\operatorname{crit}}} \\
}$$ is a Cartesian diagram of sheaves of rings on $X'$ (where Cartesian for sheaves means Cartesian on the stalks). The functor ${\mathbb{G}}_m$ from the category of rings with $1$ to the category of abelian groups that associates to a ring its group of units, commutes with limits and filtered colimits (e.g. with taking stalks). This implies that the diagram $$\xymatrix{
{\mathbb{G}}_{m,X'} \ar[r]\ar[d] & \nu'_*{\mathbb{G}}_{m,\wtl{X}} \ar[d] \\
i'_*{\mathbb{G}}_{m,X_{\operatorname{sing}}} \ar[r] & \nu'_*j_*{\mathbb{G}}_{m,\wtl{X}_{\operatorname{crit}}} \\
}$$ is Cartesian. Hence the exactness of the sequence $$0 \lrar {\mathbb{G}}_{m,X'} \lrar \nu'_*{\mathbb{G}}_{m,\wtl{X}} \oplus i'_*{\mathbb{G}}_{m,X_{\operatorname{sing}}} \lrar \nu'_*j_*{\mathbb{G}}_{m,\wtl{X}_{\operatorname{crit}}}.$$ It remains to check that the last map here is surjective. The morphism $\nu$ is finite, hence the functor $\nu_*$ is exact [@EC Cor. II.3.6]. The map ${\mathbb{G}}_{m,\wtl{X}} \to j_*{\mathbb{G}}_{m,\wtl{X}_{\operatorname{crit}}}$ is surjective, because $j:\wtl{X}_{\operatorname{crit}}\to \wtl{X}$ is a closed embedding, and thus $\nu'_*{\mathbb{G}}_{m,\wtl{X}} \to \nu'_*j_*{\mathbb{G}}_{m,\wtl{X}_{\operatorname{crit}}}$ is surjective too.

For fields $k_1,\ldots,k_n$, we have $\operatorname{Br}\big(\coprod_{i=1}^n\operatorname{Spec}(k_i)\big)=\oplus_{i=1}^n\operatorname{Br}(k_i)$.

\[p1\] Let $C$ be a reduced, geometrically connected, projective curve, and let $\Lambda$, $\Pi$ and $\Psi$ be the schemes defined 