isomorphisms $$\wtl C\setminus j(\Psi)\tilde\lrar C'\setminus i'(\Pi) 
\tilde\lrar C\setminus i(\Pi).$$

An algebraic group over $\Pi$ is a product $G=\prod_\pi i_{\pi*}(G_\pi)$, where $\pi$ ranges over the irreducible components of $\Pi$, $i_\pi: \operatorname{Spec}(k(\pi))\to\Pi$ is the natural closed immersion, and $G_\pi$ is an algebraic group over the field $k(\pi)$.

\[exact\] $(i)$ The canonical maps ${\mathbb{G}}_{m,C'}\to \nu'_*{\mathbb{G}}_{m,\wtl C}$ and ${\mathbb{G}}_{m,C'}\to i'_*{\mathbb{G}}_{m,\Pi}$ give rise to the exact sequence of étale sheaves on $C'$ $$0\to{\mathbb{G}}_{m,C'}\to \nu'_*{\mathbb{G}}_{m,\wtl C}\oplus i'_*{\mathbb{G}}_{m,\Pi}\to i'_*\nu'_*{\mathbb{G}}_{m,\Psi}
\to 0, \label{ee1}$$ where $\nu'_*{\mathbb{G}}_{m,\Psi}$ is an algebraic torus over $\Pi$.

$(ii)$ The canonical map ${\mathbb{G}}_{m,C}\to \nu''_*{\mathbb{G}}_{m,C'}$ gives rise to the exact sequence of étale sheaves on $C$: $$0\to{\mathbb{G}}_{m,C}\to \nu''_*{\mathbb{G}}_{m,C'}\to i_*{\mcal{U}}\to 0, \label{ee2}$$ where ${\mcal{U}}$ is a commutative unipotent group over $\Pi$.

This is essentially well known, see [@BLR], the proofs of Propositions 9.2.9 and 9.2.10, or [@Liu Lemma 7.5.12]. By [@EC Thm. II.2.15 (b), (c)] it is enough to check the exactness of (\[ee1\]) at each geometric point $\bar x$ of $C'$. If $\bar x\notin i'(\Pi)$, this is obvious since locally at $\bar x$ the morphism $\nu'$ is an isomorphism, and the stalks $(i'_*{\mathbb{G}}_{m,\Pi})_{\bar x}$ and $(i'_*\nu'_*{\mathbb{G}}_{m,\Psi})_{\bar x}$ are zero. Now let $\bar x\in i'(\Pi)$, and let $\O_{\bar x}$ be the strict henselisation of the local ring 