Let $k$ be an algebraically closed field, $C$ a proper reduced connected scheme over $k$ of dimension 1, whose singularity is at worse ordinary, $\pi : \tilde{C} \to C$ the normalization of $C$ and $D_i$ be the irreducible components of $\tilde{C}$.
And write the structure morphism $f:C \to k, g : \tilde{C} \to k$.
Then, since "group algebraic spaces of locally of finite type" over a filed are schemes, its Picard functor is representable by a scheme, and denote its identity component by $J$.
Since $C$ is a curve, $J$ is smooth, so is an algebraic group.
How can I show that $J$ is semi abelian?
Here is what I have tried:
This is written in ch.9.2, proposition 10 in Bosch et. al.'s Neron Models.
First, write $r = (\text{the number of irreducible components of } C)$, $C_{\text{sing}} = \{ x_i \}_{i = 1, \dots, N}$, $ \pi^{-1}(x_i) = \{ x_{ij} \}_{j = 1, \dots, n_i}$ and $M$ the rank of $H^1(\Gamma, \mathbb{Z})$, where $\Gamma$ is the graph associated with $C$.
We have a short exact (S) $ 1 \to \mathscr{O}_C^* \to \pi_*\mathscr{O}_\tilde{C}^* \to \mathscr{Q} \to 1 $, where $\mathscr{Q} = \oplus_i k_{x_i}^{n_i - 1}$, where $k_x$ is the skyscraper sheaf at $x \in C$ associated with $k$.
Then the author says that we have the long exact
$$ 1 \to f_* \mathbb{G}_{m, C} \to f_* \pi_* \mathbb{G}_{m, \tilde{C}} \to f_* \mathscr{Q} \to R^1 f_* \mathbb{G}_{m, C} \to R^1 (f_* \pi_*) \mathbb{G}_{m, \tilde{C}} \to 1 $$
in the big etale topology on $\operatorname{Spec}k$.
I don't understand why.
(And I don't know what is $\mathscr{Q}$ as an etale sheaf.
A quasi-coherent sheaf of module (in the usual sense) induces the big etale sheaf.
But now $\mathscr{Q}$ is not a quasi-coherent module.)
I think there exists the "big etale version" of (S), $ 1 \to \mathbb{G}_{m, C} \to \pi_* \mathbb{G}_{m, \tilde{C}} \to \mathscr{C} \to 1 $ on the big etale topology of $C$, where
$\mathscr{C} = \oplus_i i_{x_i *} \mathbb{G}_{m, k}^{n_i - 1}$, $i_x$ is the canonical morphism $\{x\} \to C$.
If so, then since $R^1 f_* \mathbb{G}_{m, C} = \operatorname{Pic}_C, (R^1 f_*) \circ \pi_* = R^1 g_*, R^1f_* \mathscr{C} = 0$ and $f_* \mathscr{C} = \mathbb{G}_{m, k}^M$, we have
$$ 0 \to \mathbb{G}_m \to \mathbb{G}_m^r \to \mathbb{G}_m^M \to \operatorname{Pic}_{C/k} \to \operatorname{Pic}_{\tilde{C}/k} \to 0$$
on the big etale topology on $\operatorname{Spec}k$, hence ok.
How can I show it?
I could show it for the standard Neron n-gon (over a general scheme $S$). (For the definition of Neron polygon, see the paper of Deligne-Rapoport.) But I could not show it for general stable curves.
Any other proof will be appreciated, but if possible, I want online-available references.
Thank you very much!
