Let $\bar x$ be a geometric point of $i(\Pi)$, and let $\bar y$ be the unique geometric point of $C'$ such that $\nu''(\bar y)=\bar x$. Let $\O_{\bar x}$ and $\O_{\bar y}$ be the corresponding strictly henselian local rings. The stalk $(\nu''_*{\mathbb{G}}_{m,C'}/{\mathbb{G}}_{m,C})_{\bar x}$ is $\O_{\bar y}^*/\O_{\bar x}^*$, and according to [@Liu Lemma 7.5.12 (c)], this is a unipotent group over the field $k(\bar x)$. This finishes the proof.

The first part of Proposition \[exact\] has an alternative proof which is easier to generalise to higher dimension. Let $X$ be a projective $k$-variety with normalisation morphism $\nu: \wtl{X} \to X$. Assume that $\wtl{X}$, $X_{\operatorname{sing}}$ and $\wtl{X}_{\operatorname{crit}}$ are smooth, where $X_{\operatorname{sing}}$ is the singular locus of $X$ and $\wtl{X}_{\operatorname{crit}} = \nu^{-1}(X_{\operatorname{sing}}) \subseteq \wtl{X}$ is the critical locus of $\nu$. (This assumption is automatically satisfied when $X$ is a curve.) The analogue of $C'$ is the $K$-variety $X'$ given by the pushout in the square $$\xymatrix{
\wtl{X}_{\operatorname{crit}} \ar^{j}[r]\ar_{g}[d] & \wtl{X} \ar^{\nu'}[d] \\
X_{\operatorname{sing}} \ar^{i'}[r] & X' \\
}$$ This pushout exists in the category of $K$-varieties since $i'$ is a closed embedding and $g$ is an affine morphism of smooth projective varieties (see [@FE Thm. $5.4$]). One then proves that the sequence of sheaves 