1.1 Preliminaries

In this section we collect together properties of Néron models and Weil restriction of scalars. Most of these may be found in [Reference Bosch, Lütkebohmert and RaynaudBLR90], especially Chapter 10.

Recall that if $G/F$ is a smooth group scheme of finite type, then a Néron model for $G$ is a smooth separated group scheme $\mathcal {G}/S$ with generic fibre $G$, such that for every smooth $S$-scheme $S'$, the canonical map $\mathcal {G}(S') \to \mathcal {G}(S'_F)=G(S'_F)$ is bijective. If $\mathcal {G}$ exists, it is unique up to unique isomorphism. (In [Reference Bosch, Lütkebohmert and RaynaudBLR90] these are called Néron lft-models.) The identity component $\mathcal {G}^0$ of $\mathcal {G}$ is a smooth group scheme of finite type. The formation of Néron models commutes with strict henselization and completion of the base ring $R$. If $G \otimes _F \widehat {F}^{\mathrm {sh}}$ does not contain a copy of $\mathbb {G}_{\mathrm a}$, then $G$ has a Néron model [Reference Bosch, Lütkebohmert and RaynaudBLR90, 10.2 Thm.2]. (More generally, this holds if $S$ is merely a semilocal Dedekind scheme.) We write $\Phi (G)$ for the component group $(\mathcal {G}_s/\mathcal {G}_s^0)(k^\mathrm {sep})$. If $k$ is perfect, then by Chevalley's theorem [Reference ConradCon02] $\mathcal {G}_s$ has a unique maximal connected affine smooth subgroup scheme $\mathcal {G}_s^{0,\mathrm {lin}}$, and we then write $\mathbb {X}(G)$ for the character group $\operatorname {Hom}(\mathcal {G}_s^{0,\mathrm {lin}}\otimes _k\bar k,\mathbb {G}_{\mathrm {m}})$, a finite free $\mathbb {Z}$-module with a continuous action of $\operatorname {Gal}(\bar k/k)$.

Let $0 \to G_1 \to G_2 \to G_3 \to 0$ be an exact sequence of smooth connected $F$-groups which have Néron models $\mathcal {G}_i$. Consider the complexes

(1.1.1)\begin{gather} 0 \to \mathcal{G}_1 \to \mathcal{G}_2 \to \mathcal{G}_3 \to 0, \end{gather}

(1.1.2)\begin{gather}0 \to \mathcal{G}_1^0 \to \mathcal{G}_2^0 \to \mathcal{G}_3^0 \to 0, \end{gather}

(1.1.3)\begin{gather}0 \to \Phi(G_1) \to \Phi(G_2) \to \Phi(G_3) \to 0. \end{gather}

The following two exactness results are a restatement of [Reference ChaiCha00, Remark (4.8)(a)], with the same proof, which we give for the reader's convenience.

Proof. (a) Since locally for the smooth topology the morphism $\mathcal {G}_2 \to \mathcal {G}_3$ of group schemes has a section, it is evidently surjective. Let $\mathcal {G}'$ denote its kernel. By [Reference Liu and LorenziniLL01, Lemma 4.3(b)], $\mathcal {G}'$ is smooth. The canonical morphism $\mathcal {G}_1 \to \mathcal {G}_2$ factors though a morphism $\gamma \colon \mathcal {G}_1 \to \mathcal {G}'$ which is the identity on generic fibres, and since $\mathcal {G}_1$ is a Néron model, there is a morphism $\delta \colon \mathcal {G}' \to \mathcal {G}_1$ which is the identity on generic fibres. As $\mathcal {G}_1$ and $\mathcal {G}'$ are separated over $S$, $\gamma$ and $\delta$ are mutually inverse isomorphisms.

(b) The map $\mathcal {G}_2^0 \to \mathcal {G}_3^0$ is surjective, so we have an exact sequence

\[ 0 \to \mathcal{G}_1 \cap \mathcal{G}_2^0 \to \mathcal{G}_2^0 \to \mathcal{G}_3^0 \to 0 \]

in which each term is of finite type over $S$. Hence, $\mathcal {G}_{1,s}^0$ has finite index in $\mathcal {G}_{1,s} \cap \mathcal {G}_{2,s}^0$, and since $\Phi (G_1)$ is torsion-free we have $\mathcal {G}_1 \cap \mathcal {G}_2^0 = \mathcal {G}_1^0$. Thus, (1.1.2) and therefore also (1.1.3) are exact.

We will need the following minor generalization of a result from [Reference Bosch, Lütkebohmert and RaynaudBLR90].

Proposition 1.3 Let

\[ 0 \to \mathcal{G}_1 \to \mathcal{G}_2 \to \mathcal{G}_3 \to 0 \]

be an exact sequence of smooth $S$-group schemes. If $\mathcal {G}_1$ and $\mathcal {G}_3$ are the Néron models of their generic fibres, the same is true for $\mathcal {G}_2$.

From [Reference Bosch, Lütkebohmert and RaynaudBLR90, § 7.6] we recall basic properties of Weil restriction. Let $Z'/Z$ be a finite flat morphism of finite presentation. If $Y$ is a quasiprojective $Z'$-scheme, then the Weil restriction $\mathcal {R}_{Z'/Z} Y$ exists, and is characterized by its functor of points $\mathcal {R}_{Z'/Z}Y(-) = Y(- \times _Z Z')$. If $Y$ is smooth over $Z'$, then $\mathcal {R}_{Z'/Z} Y$ is smooth over $Z$. If $Y \to X$ is a closed immersion of quasiprojective $Z'$-schemes, then $\mathcal {R}_{Z'/Z} Y \to \mathcal {R}_{Z'/Z} X$ is a closed immersion. If $Z' \to Z$ is surjective and $Y$ is a quasiprojective $Z$-scheme, then the canonical map $Y \to \mathcal {R}_{Z'/Z} (Y \times _Z Z')$ is a closed immersion.

Now let $k$ be a field, $k'$ a finite $k$-algebra, and $k''$ a finite flat $k'$-algebra. Let $Y$ be a quasiprojective $k$-scheme. There is then a canonical map

\[ g \colon \mathcal{R}_{k'/k} (Y \otimes_k k' ) \to \mathcal{R}_{k''/k} (Y \otimes_k k''). \]

We may write $k' = k_1' \times k_2'$ where $\operatorname {Spec} k_1'\subset \operatorname {Spec} k'$ is the image of $\operatorname {Spec} k''$ (and $k_2'$ is possibly zero). The morphism $g$ then factors

\begin{align*} \mathcal{R}_{k'/k} (Y \otimes_k k' ) &= \mathcal{R}_{k_1'/k} (Y \otimes_k k_1' ) \times_{\operatorname{Spec} k} \mathcal{R}_{k_2'/k} (Y \otimes_k k_2') \\ &\xrightarrow{\mathrm{pr}_1}\mathcal{R}_{k_1'/k} (Y \otimes_k k_1' ) \to \mathcal{R}_{k_1'/k} \mathcal{R}_{k''/k_1'} (Y \otimes_k k'' ) = \mathcal{R}_{k''/k} (Y \otimes_k k'' ) \end{align*}

and the second arrow is a closed immersion. In particular, if $Y$ is a smooth $k$-group, then $g$ is a surjection onto a closed subgroup scheme, and its cokernel is smooth.

Let $k$ be a field and $k'$ a finite $k$-algebra. Then $\mathcal {R}_{k'/k}\mathbb {G}_{\mathrm {m}}$ is a connected smooth $k$-group scheme of finite type. It is a torus if and only if $k'/k$ is étale.

We return to Néron models. Recall that the multiplicative group $\mathbb {G}_{\mathrm {m}}/F$ has a Néron model $\mathcal {G}_\mathrm {m} /S$, whose special fibre is $\mathbb {G}_{\mathrm {m}} \times \mathbb {Z}$. It fits into an exact sequence of group schemes

\[ 0 \to \mathbb{G}_{\mathrm{m}} \to \mathcal{G}_\mathrm{m} \xrightarrow{v_F} s_*\mathbb{Z} \to 0, \]

where on $R$-points $v_F$ is the normalized valuation $v_F \colon \mathcal {G}_\mathrm {m} (R) = F^\times \to \mkern -15mu\to \mathbb {Z}$.

Let $F'$ be a finite étale $F$-algebra, $R' \subset F'$ the normalization of $R$ in $F'$, $S' = \operatorname {Spec} R'$. Let $F' \otimes _F F^{\mathrm {sh}} = \prod _{i\in I} F_i$, where the fields $F_i$ are totally ramified extensions of $F^{\mathrm {sh}}$, of degrees $e_i p^{s_i}$, where $p^{s_i}$ is the degree of the (purely inseparable) residue class extension.

Here we use $\mathcal {G}_\mathrm {m}$ to denote also the Néron model of $\mathbb {G}_{\mathrm {m}}$ over the semilocal base $S'$.

Proof. (a) The first statement follows from [Reference Bosch, Lütkebohmert and RaynaudBLR90, Propositions 10.1/4 and 6]. For the second, replacing $F$ by $F^{\mathrm {sh}}$ we are reduced to the case of a totally ramified field extension $F'/F$. Then as $\mathcal {G}_{{\mathrm m},s}\simeq \mathbb {G}_{\mathrm m,s}\times \mathbb {Z}$, we have $(\mathcal {R}_{S'/S}\mathcal {G}_\mathrm {m})_s = \mathcal {R}_{R'\otimes k/k}\mathbb {G}_{\mathrm {m}} \times \mathcal {R}_{R'\otimes k/k}\mathbb {Z}$. As the first factor is connected, and the second is $\mathbb {Z}$ (since $R'\otimes k/k$ is radicial) we get $\Phi (\mathcal {R}_{F'/F}\mathbb {G}_{\mathrm {m}})\simeq \mathbb {Z}$, and the fact that this isomorphism is given by the valuation follows from [Reference Bosch, Lütkebohmert and RaynaudBLR90, 1.1/Proposition 7].

(b) By Corollary 1.2, the exact sequence $0\to \mathbb {G}_{\mathrm {m}}\to \mathcal {R}_{F'/F}\mathbb {G}_{\mathrm {m}} \to (\mathcal {R}_{F'/F}\mathbb {G}_{\mathrm {m}})/\mathbb {G}_{\mathrm {m}}\to 0$ gives rise to exact sequences of Néron models and component groups. It is therefore enough to show that the map $\Phi (\mathbb {G}_{\mathrm {m}})=\mathbb {Z} \to \Phi (\mathcal {R}_{F'/F}\mathbb {G}_{\mathrm {m}})=\mathbb {Z}^I$ is equal to $e$. Replacing $F$ by $F^{\mathrm {sh}}$ again, we are reduced to the case when $F'/F$ is a totally ramified field extension of degree $ep^s$ with residue degree $p^s$. Then by part (a) we have a commutative square

proving the result.

1.2 Graphs and Picard schemes of singular curves

In this section we work over an arbitrary field $k$. By a curve over $k$ we shall mean a $k$-scheme $X$ of finite type which is equidimensional of dimension 1 and Cohen–Macaulay (i.e. has no embedded points). Let $\{X_j\}$ be the irreducible components of $X$, and $\eta _j$ the generic point of $X_j$. The local ring $\mathcal {O}_{X,\eta _j}$ is Artinian, and following Raynaud [Reference RaynaudRay70, (6.1.1) and (8.1.1)] we write $d_j$ for its length, and $\delta _j$ for the total multiplicity of $X_j$ in $X$. If $k'/k$ is a radicial closure of $k$, and $\eta '_j\in X\otimes k'$ is the point lying over $\eta _j$, then $\delta _j$ equals the length of the local ring of $\eta '_j$. Moreover, $\delta _j = d_j[\kappa (\eta _j)\cap k' : k]=d_jp^{n_j}$ for some $n_j\ge 0$.

Until the end of this section, $k$ denotes an algebraically closed field. We review the well-known description of the toric part of the Picard scheme of a singular curve over $k$.

Let $Y/k$ be a reduced proper curve, and $Y^\mathrm {sing}\subset Y(k)$ its set of singular points. Write $\phi \colon \widetilde Y \to Y$ for its normalization. Define sets

\[ A=Y^\mathrm{sing}\subset Y(k),\quad B=\phi^{-1}(Y^\mathrm{sing})\subset \tilde Y(k),\quad C=\pi_0(\widetilde Y). \]

We have maps

\begin{align*} \phi \colon B \to A,\quad \psi\colon B \to C, \end{align*}

where $\psi$ maps $x\in B$ to the connected component of $\widetilde Y$ containing it.

The extended graph $\widetilde{\Gamma} _Y = (\widetilde V,\widetilde E)$ of $Y$ is the graph with vertices $\widetilde V$ and edges $\widetilde E$, where:

• – $\widetilde V= A\sqcup C$, $\widetilde E = B$;

• – the endpoints of an edge $b\in B$ are $\phi (b)\in A$ and $\psi (b)\in C$.

The graph $\widetilde{\Gamma} _Y$ is bipartite, and therefore has a canonical structure of directed graph, by directing the edge $b$ so that its source is $\phi (b)$.

Suppose $Y$ only has double points (meaning that if $y\in Y^\mathrm {sing}$ then $\phi ^{-1}(y)$ has exactly two elements). The reduced graph $\Gamma _Y=(V,E)$ is the undirected graph (possibly with multiple edges and loops) whose vertex set is $V=\pi _0(\widetilde Y)$ and edge set is $E=Y^\mathrm {sing}$. It is obtained from $\tilde \Gamma _Y$ by, for each vertex $v\in A$, deleting $v$ and replacing the two edges incident to $v$ with a single edge. There is a canonical homeomorphism between the geometric realizations of $\widetilde{\Gamma} _Y$ and $\Gamma _Y$, under which $v\in A$ is mapped to the midpoint of the replacing edge. If $Y/k$ is a proper curve, not necessarily reduced, we define $\widetilde{\Gamma} _Y=\widetilde{\Gamma} _{Y^\mathrm {red}}$, $\Gamma _Y=\Gamma _{Y^\mathrm {red}}$, where $Y^\mathrm {red}\subset Y$ is the reduced subscheme.

Let $G=\operatorname {Pic}^0_Y$ be the identity component of the Picard scheme of $Y$. It is a smooth group scheme of finite type over $k$, classifying line bundles on $Y$ whose restriction to each irreducible component has degree zero. The filtration of $G$ by its linear and unipotent subgroups is described as follows.

Let $Y'\to Y$ be the ‘seminormalization’ of $Y$, which is obtained from $Y$ by replacing its singularities with singularities which are étale locally isomorphic to the union of coordinate axes in $\mathbb {A}^N_k$. The normalization map factors into a pair of finite morphisms $\widetilde Y \overset {\phi '}\to Y' \to Y$. These give rise to a commutative diagram, whose rows are exact:

(where $G^\mathrm {unip}$ is the maximal connected unipotent subgroup of $G$) giving an isomorphism $\ker \phi ^{\prime *}\simeq G^\mathrm {tor}=G^\mathrm {lin}/G^\mathrm {unip}$ by the snake lemma.

To give a line bundle on $Y'$ is equivalent to giving a line bundle on $\widetilde Y$ together with descent data for $\phi '\colon \widetilde Y\to Y'$, so the toric part $G^\mathrm {tor}$ classifies trivial line bundles on $\widetilde Y$ equipped with descent data to $Y'$. For the trivial bundle $\mathcal {O}_{\widetilde Y}$, to give such descent data is equivalent to giving, for each singular point $y\in Y$, an element of $(k^\times )^{\phi ^{-1}(y)}/k^\times$. The automorphism group of $\mathcal {O}_{\widetilde Y}$ is $(k^\times )^{\pi _0(\widetilde Y)}$. Hence, $G^\mathrm {tor}$ is canonically

\[ \mathbb{G}_{\mathrm{m}}^{\pi_0(\widetilde Y)} \backslash \prod_{y\in Y^{\mathrm{sing}}} \big(\mathbb{G}_{\mathrm{m}}^{\phi^{-1}(y)}/\mathrm{diag}(\mathbb{G}_{\mathrm{m}})\big). \]

Here $\mathbb {G}_{\mathrm {m}}^{\pi _0(\widetilde Y)}$ acts on $\mathbb {G}_{\mathrm {m}}^{\phi ^{-1}(y)}$ by the dual of the map $\phi ^{-1}(y)\subset \widetilde Y(k)\overset \psi \to \pi _0(Y)$ associating to $x\in \widetilde Y$ the connected component of $\widetilde Y$ containing it. The character group of $G^\mathrm {tor}$ is therefore the kernel of the map

\[ \mathbb{Z}[B] \xrightarrow{(\psi,\phi)} \mathbb{Z}[C]\oplus\mathbb{Z}[A], \]

which (after replacing $\phi$ with $-\phi$) is the chain complex of $\widetilde{\Gamma} _Y$. This gives the formula [Reference Deligne and RapoportDR73, I.3]

(1.2.1)\begin{equation} \operatorname{Hom}(G^\mathrm{lin} ,\mathbb{G}_{\mathrm{m}}) = H_1(\widetilde\Gamma_Y,\mathbb{Z}). \end{equation}

Suppose now that $Y$ is a proper curve over $k$, not necessarily reduced. The map $\operatorname {Pic}^0_Y \to \operatorname {Pic}^0_{Y^\mathrm {red}}$ is an epimorphism, and its kernel is a connected unipotent group scheme, so (1.2.1) remains valid.

If $k$ is merely assumed to be perfect, (1.2.1) holds as an isomorphism of $\operatorname {Gal}(\bar k/k)$-modules.

If $Y$ only has double points with distinct branches, then by the homeomorphism $\widetilde{\Gamma} _Y\to \Gamma _Y$ we obtain the formula

(1.2.2)\begin{equation} \operatorname{Hom}(G^\mathrm{lin} ,\mathbb{G}_{\mathrm{m}}) = H_1(\Gamma_Y,\mathbb{Z}). \end{equation}

1.3 The Néron model of $J$

In preparation for § 1.6, we review in more detail the results of Raynaud. We will follow mainly the notation of [Reference RaynaudRay70] (see also [Reference Bosch, Lütkebohmert and RaynaudBLR90], where the notation is slightly different).

We consider a proper flat morphism $\mathcal {X} \to S=\operatorname {Spec} R$, satisfying the following hypotheses (H1)–(H3).

1. (H1) The generic fibre $X := \mathcal {X}_F$ is a smooth geometrically connected curve over $F$ (in particular, $\Gamma (\mathcal {X},\mathcal {O}_\mathcal {X})=R$).

2. (H2) The scheme $\mathcal {X}$ is regular.

Let the irreducible components of $\mathcal {X}_s$ be indexed by the set $C$, and for $j\in C$, let $\mathcal {X}_j\subset \mathcal {X}_s$ be the scheme-theoretic closure of the corresponding maximal point of $\mathcal {X}_s$, $\delta _j=p^{n_j}d_j$ its total multiplicity (§ 1.2), and $Y_j= \mathcal {X}_j^\mathrm {red}$. Define $\delta =\gcd \{\delta _j\}$, $d=\gcd \{d_j\}$.

1. (H3) We have $(\delta,p)=1$.

Hypotheses (H1) and (H2) imply that Raynaud's condition $(\mathrm N)^*$ is satisfied [Reference RaynaudRay70, (6.1.4)]. Hypothesis (H1) is not particularly restrictive, since one may always reduce to this case using Stein factorization. In the presence of hypotheses (H1) and (H2), hypothesis (H3) implies that $\mathcal {X}/S$ is cohomologically flat (equivalently, that $\Gamma (\mathcal {X}_s,\mathcal {O}_{\mathcal {X}_s})=k$), by [Reference RaynaudRay70, (7.2.1)].

Let $J=\operatorname {Pic}^0_{X/F}$ be the Jacobian variety of $X$, and let $\mathcal {J}$ be the Néron model of $J$.

The relative Picard functor $P=\operatorname {Pic}_{\mathcal {X}/S}$ is the sheafification (for the fppf topology) of the functor on the category of $S$-schemes

\[ S' \mapsto \operatorname{Pic}(\mathcal{X}\times_S S'). \]

There is a morphism of abelian sheaves $\deg \colon P \to \mathbb {Z}$ which takes a line bundle to its total degree along the fibres, and $P'\subset P$ denotes its kernel. By [Reference RaynaudRay70, (5.2) and (2.3.2)], $P$ and $P'$ are formally smooth algebraic spaces over $S$, and the closure $E\subset P$ of the zero section is an étale algebraic space over $S$, contained in $P'$. The maximal separated quotient $Q=P/E$ is a smooth separated $S$-group scheme, and the subgroup $Q'=P'/E$ is the closure in $Q$ of the identity component $Q^0$ (proof of [Reference RaynaudRay70, (8.1.2)(iii)]). One also has the subgroup $Q^\tau \subset Q$, which is the inverse image of the torsion subgroup of $Q/Q^0$. As $\mathcal {X}$ is regular, condition d) of [Reference RaynaudRay70, (8.1.2)] holds, and so $Q^\tau$ is closed in $Q$. By definition $\deg (Q^\tau )=0$, and therefore $Q'=Q^\tau$. Therefore, [Reference RaynaudRay70, (8.1.2) and (8.1.4)(b)] imply that $\mathcal {J}=Q'=P'/E$.

(If hypothesis (H3) is not satisfied, then $P$ is in general not representable, but it still has a maximal separated quotient $Q$ which is a smooth separated $S$-group scheme [Reference RaynaudRay70, (4.1.1)]. If moreover $k$ is perfect, then $Q'$ again equals $\mathcal {J}$ (see [Reference RaynaudRay70, (8.1.4)(a)]).)

Let $P_s^0$ be the identity component of $P_s$. We have $P_s^0=\operatorname {Pic}^0_{\mathcal {X}_s/k}$, the identity component of the Picard scheme of $\mathcal {X}_s$. By [Reference RaynaudRay70, (6.4.1)(3)], the intersection $P^0_s\cap E_s$ is a constant group scheme over $k$, cyclic of order $d$, generated by the class of the line bundle $\mathcal {L}'=\mathcal {O}(\sum _j(d_j/d)Y_j)$. (Because $\mathcal {X}$ is regular, the integers $d$ and $d'$ (see [Reference RaynaudRay70, (6.1.11)(3)]) are equal.) Therefore, $\mathcal {J}_s^0$ is canonically isomorphic to $\operatorname {Pic}^0_{\mathcal {X}_s/k}/\langle \mathcal {L}'\rangle$ and, in particular, if $d=1$, then $\mathcal {J}_s^0=\operatorname {Pic}^0_{\mathcal {X}_s/k}$.

Suppose that $k$ is perfect and $d=1$. Combining the above with the discussion in § 2.2, we then have an isomorphism of $\operatorname {Gal}(\bar k/k)$-modules

\[ \mathbb{X}(J) := \operatorname{Hom}(\mathcal{J}_s^{0,\mathrm{lin}}\otimes_k\bar k, \mathbb{G}_{\mathrm{m}}) = H_1(\tilde\Gamma_{\mathcal{X}_s\otimes\bar k},\mathbb{Z}). \]

Finally, we recall the description of the component group. First suppose that $R$ is strictly Henselian ($k$ not necessarily perfect). Then [Reference RaynaudRay70, (8.1.2)] shows that the component group $\Phi (J)=\mathcal {J}_s/\mathcal {J}_s^0$ is computed as follows: by the above, $\Phi (J)= Q'_s/Q^0_s$ is the cokernel of the map

\[ E_s \to P'_s/P^0_s =\ker(\deg\colon P_s/P^0_s \to \mathbb{Z}). \]

One has an isomorphism

\[ P_s/P_s^0 \simeq \mathbb{Z}^C,\quad (\mathcal{L}\in P_s) \mapsto (\deg \mathcal{L}|_{Y_j})_j. \]

Let $D\subset \operatorname {Div}\mathcal {X}$ be the group of Cartier divisors supported in the special fibre, and $D_0\subset D$ the subgroup of principal divisors. By [Reference RaynaudRay70, (6.1.3)] one has $E_s=D/D_0$. As $\mathcal {X}$ is regular and $R=\Gamma (\mathcal {X},\mathcal {O}_\mathcal {X})$, $D$ is freely generated by the set of reduced components $\{Y_j\}$, and $D_0$ is the subgroup generated by the divisor $(\varpi )$ of the special fibre. The complex of [Reference RaynaudRay70, (8.1.2)(i)] then becomes

(1.3.1)\begin{align} 0 \to \mathbb{Z} \xrightarrow{i} \mathbb{Z}[C] \xrightarrow{a} \mathbb{Z}^C \xrightarrow{b} \mathbb{Z} \to 0 \end{align}

where the maps are

(1.3.2)\begin{align} i(1) &= \sum_{j \in C}d_j (j) , \nonumber\\ a(\ell) &= \bigg(\frac1{\delta_j}\deg\mathcal{O}_{\mathcal{X}}(Y_\ell)|_{Y_j} \bigg)_{j\in C} = \bigg(\frac1{p^{n_j}}(Y_j.Y_\ell)\bigg)_{j\in C} \quad (\ell\in C) , \nonumber\\ b(m) &= \sum_{j\in C}\delta_jm_j, \quad{m=(m_j)\in \mathbb{Z}^C,} \end{align}

and $\Phi (J)=\ker (b)/\operatorname {im}(a)$.

If $\mathcal {X}$ is semistable (meaning that $\mathcal {X}_s$ is smooth over $k$ apart from double points with distinct tangents), then both the character group and component group can be described in terms of the reduced graph $\Gamma _{\mathcal {X}_s}$. The character group equals the homology of $\Gamma _{\mathcal {X}_s}$. The map $a\colon \mathbb {Z}[C] \to \ker (b)=\mathbb {Z}^{C,0}\subset \mathbb {Z}^C$ is, after identifying $\mathbb {Z}[C]$ with $\mathbb {Z}^C$, the Laplacian of the graph $\Gamma _{\mathcal {X}_s}$, which takes a vertex $v\in C$ to $\sum (v)- (v')\in \mathbb {Z}[C]_0$, the sum taken over all edges joining $v$ to an adjacent vertex $v'$.

In general, we have an isomorphism of $\operatorname {Gal}(\bar k/k)$-modules $\Phi (J) =\ker (b)/\operatorname {im}(a)$, where $a$, $b$ are the maps in the complex (1.3.1) for the base change $\mathcal {X}\otimes _R R^{\mathrm {sh}}$.

1.4 Generalized Picard schemes of singular curves

Let $k$ be a field, and $Y/k$ a proper curve (in the sense of § 1.2). Write $k'$ for the $k$-algebra $\Gamma (Y,\mathcal {O}_Y)$. By a generalized modulus on $Y$ we mean a morphism of $k$-schemes $\Sigma \to Y$, where $\Sigma$ is a finite $k$-scheme, flat over $\operatorname {Spec} k'$.

We define $\operatorname {Pic}_{(Y,\Sigma )/k}$ to be the scheme classifying line bundles on $Y$ together with a trivialization of the pullback to $\Sigma$. Precisely, consider the functor $\mathcal {F}$ which to a $k$-scheme $S$ associates the set of equivalence classes of pairs $(\mathcal {L},\alpha )$, where $\mathcal {L}$ is a line bundle on $Y\times S$ and $\alpha \colon \mathcal {O}_{\Sigma \times S} \xrightarrow \sim (g\times \mathrm {id}_S)^*\mathcal {L}$ is a trivialization, and where pairs $(\mathcal {L},\alpha )$, $(\mathcal {L}',\alpha ')$ are equivalent if there exists an isomorphism $\sigma \colon \mathcal {L} \xrightarrow \sim \mathcal {L}'$ such that $\alpha '=g^*(\sigma )\circ \alpha$.

Let $Y = Y_1 \sqcup Y_2$ where $g(\Sigma )$ is disjoint from $Y_2$ and meets each connected component of $Y_1$. If $Y_2 = \emptyset$, then by Lemma 1.5 we are in the situation of [Reference RaynaudRay70, § 2], and $\mathcal {F}$ is a sheaf for the fppf topology which we denote $\operatorname {Pic}_{(Y,\Sigma )/k}$. In general, we define $\operatorname {Pic}_{(Y,\Sigma )/k}$ to be the sheafification of $\mathcal {F}$ for the fppf topology. Obviously $\operatorname {Pic}_{(Y,\Sigma )/k} = \operatorname {Pic}_{(Y_1 ,\Sigma )/k} \times _k \operatorname {Pic}_{Y_2/k}$. Put $k' = k_1 \times k_2$ where $k_i = \Gamma (Y_i , \mathcal {O}_{Y_i} )$. From [Reference RaynaudRay70] we then obtain the following.

Proposition 1.6 The functor $\operatorname {Pic}_{(Y,\Sigma )/k}$ is represented by a smooth $k$-group scheme, and there is an exact sequence of smooth group schemes

(1.4.1)\begin{equation} 0 \to H \to \operatorname{Pic}_{(Y,\Sigma)/k} \to \operatorname{Pic}_{Y/k} \to 0, \end{equation}

where

\[ H = H_\Sigma := \operatorname{coker}\big( \mathcal{R}_{k'/k}(\mathbb{G}_{\mathrm{m}}) \to \mathcal{R}_{\Sigma/k}(\mathbb{G}_{\mathrm{m}})\big). \]

Proof. From (2.1.2), (2.4.1), and (2.4.3) of [Reference RaynaudRay70] we get the representability of $\operatorname {Pic}_{(Y_1 ,\Sigma )/k}$ along with an exact sequence

\[ 0 \to \mathcal{R}_{k_1/k} \mathbb{G}_{\mathrm{m}} \to \mathcal{R}_{\Sigma/k} \mathbb{G}_{\mathrm{m}} \to \operatorname{Pic}_{(Y_1,\Sigma)/k} \to \operatorname{Pic}_{Y_1/k} \to 0 \]

of smooth group schemes (since, in this setting, Raynaud's $\Gamma _X^*$ and $\Gamma _R^*$ are just $\mathcal {R}_{k_1/k} \mathbb {G}_{\mathrm {m}}$ and $\mathcal {R}_{\Sigma /k}\mathbb {G}_{\mathrm {m}}$). By § 1.1, the quotient $H$ is a smooth group scheme, and taking products with $\operatorname {Pic}_{Y_2/k}$ gives the result.

Let $\operatorname {Pic}_{(Y,\Sigma )/k}^0$ denote the inverse image of $\operatorname {Pic}_{Y/k}^0$ (classifying line bundles which are of degree zero on every irreducible component of $Y$). Then $\operatorname {Pic}_{(Y,\Sigma )/k}^0$ is a smooth connected $k$-group scheme of finite type.

Let $k$ be perfect. Then $\operatorname {Pic}_{(Y,\Sigma )/k}^0$ has a maximal connected affine subgroup $\operatorname {Pic}_{(Y,\Sigma )/k}^{0,\mathrm {lin}}$ which is a linear group, and its character group has the following combinatorial description, generalizing § 1.2.

First suppose that $k$ is algebraically closed, and that $Y$, $\Sigma$ are reduced. As in § 1.2, let $\phi \colon \widetilde Y \to Y$ be the normalization, and define $A=Y^\mathrm {sing}$, $B=\phi ^{-1}(A)$, $C=\pi _0(\widetilde Y)$. Decompose $\Sigma =\Sigma ^\mathrm {sing}\sqcup \Sigma ^\mathrm {reg}$, where $z\in \Sigma ^\mathrm {sing}$ (respectively, $\Sigma ^\mathrm {reg}$) if $g(z)$ is a singular (respectively, smooth) point of $Y$. There are maps

where $\phi$, $\psi$ are as before, $\lambda$ is the restriction of $g$ to $\Sigma ^\mathrm {sing}$, and $\theta (z)$ is the component of $\widetilde Y$ containing $g(z)$.

Define the extended graph of $(Y,\Sigma )$ to be the directed graph $\widetilde{\Gamma} _{Y,\Sigma }$ obtained by adding to the graph $\widetilde{\Gamma} _Y$:

• – a single vertex $v_0$;

• – for each $z\in \Sigma ^\mathrm {sing}$, an edge from $v_0$ to the vertex $\lambda (z)\in A\subset V(\widetilde{\Gamma} _Y)$;

• – for each $z\in \Sigma ^\mathrm {reg}$, an edge from $v_0$ to the vertex $\theta (z)\in C\subset V(\widetilde{\Gamma} _Y)$.

If $Y$ only has double points and $\Sigma =\Sigma ^\mathrm {reg}$, then we may likewise define the reduced graph $\Gamma _{Y,\Sigma }$, which is the undirected graph obtained by adding to $\Gamma _Y$ a single vertex $v_0$ and, for each $z\in \Sigma$, an edge joining $v_0$ to $\theta (z)\in C=V(\Gamma _Y)$. As before, the geometric realizations of $\widetilde{\Gamma} _{Y,\Sigma }$ and $\Gamma _{Y,\Sigma }$ are canonically homeomorphic.

For arbitrary perfect $k$ and proper curve $Y$, we define $\widetilde{\Gamma} _{Y,\Sigma }$, $\Gamma _{Y,\Sigma }$ to be the graphs attached to the curve with modulus $(Y^\mathrm {red}\otimes \bar k, \Sigma ^\mathrm {red}\otimes \bar k)$, which are graphs with a continuous action of $\operatorname {Gal}(\bar k/k)$.

Proof. We may assume that $k$ is algebraically closed; the Galois equivariance of the isomorphisms will be clear from the construction. By the homeomorphism between the extended and reduced graphs, it suffices to prove part (a).

The map $g^\mathrm {red}\colon \Sigma ^\mathrm {red} \to Y^\mathrm {red}$ is a reduced modulus, and the obvious morphism induced by pullback

\[ \operatorname{Pic}_{(Y,\Sigma)/k}^0 \to \operatorname{Pic}_{(Y^\mathrm{red},\Sigma^\mathrm{red})/k}^0 \]

has unipotent kernel, since the same is true for the maps $\mathcal {R}_{\Sigma /k}\mathbb {G}_{\mathrm {m}}\to \mathcal {R}_{\Sigma ^\mathrm {red}/k}\mathbb {G}_{\mathrm {m}}$ and $\operatorname {Pic}_{Y/k}^0 \to \operatorname {Pic}_{Y^\mathrm {red}/k}^0$. Thus, the character group of $\operatorname {Pic}_{(Y,\Sigma )/k}^{0,\mathrm {lin}}$ is unchanged by passing to reduced subschemes; hence, we may assume that both $Y$ and $\Sigma$ are reduced. Next, let $Y'\to Y$ be the seminormalization. Then as $\Sigma$ is reduced, $\Sigma \to Y$ factors uniquely through $Y'$, and the resulting map

\[ \operatorname{Pic}_{(Y,\Sigma)/k}^0 \to \operatorname{Pic}_{(Y',\Sigma)/k}^0 \]

has unipotent kernel. Therefore, we may assume in addition that $Y$ is seminormal. Finally, normalization induces an exact sequence

(1.4.2)\begin{equation} 0 \to G \to \operatorname{Pic}_{(Y,\Sigma)/k}^0 \xrightarrow{\phi^*} \operatorname{Pic}_{\tilde Y/k}^0 \to 0 \end{equation}

where $G$ classifies equivalence classes of pairs $(\mathcal {L},\beta )$, where $\mathcal {L}$ is a line bundle on $Y$ whose pullback to $\tilde Y$ is trivial, and $\beta$ is a trivialization of the pullback of $\mathcal {L}$ to $\Sigma$. There is a surjective map

(1.4.3)\begin{equation} \mathbb{G}_{\mathrm{m}}^B \times \mathbb{G}_{\mathrm{m}}^{\Sigma^\mathrm{sing}} \times \mathbb{G}_{\mathrm{m}}^{\Sigma^\mathrm{reg}} \to G \end{equation}

given as follows: a tuple

\[ ((a_x)_{x\in B},(b_z)_{z\in \Sigma^\mathrm{sing}}, (c_z)_{z\in \Sigma^\mathrm{reg}})\in (\mathbb{G}_{\mathrm{m}}^B \times \mathbb{G}_{\mathrm{m}}^{\Sigma^\mathrm{sing}} \times \mathbb{G}_{\mathrm{m}}^{\Sigma^\mathrm{reg}})(k) \]

determines:

1. (i) for every $y\in A$, and any $x$, $x'\in \phi ^{-1}(y)$, isomorphisms $a_x^{-1}a_{x'} \colon x^*\mathcal {O}_{\tilde Y}=k \xrightarrow \sim k=x^{\prime *}\mathcal {O}_{\tilde Y}$ satisfying the cocycle condition and, thus, a descent of $\mathcal {O}_{\tilde Y}$ to a line bundle $\mathcal {L}$ on $Y$;

2. (ii) for every $z\in \Sigma ^\mathrm {sing}$, and every $x\in \phi ^{-1}(g(z))$, a trivialization $b_za_x^{-1}\colon k\xrightarrow \sim k=x^*\mathcal {O}_{\tilde Y}$; these trivializations are compatible with the descent data (i) and therefore give trivializations $k\xrightarrow \sim z^*\mathcal {L}$ for every $z\in \Sigma ^\mathrm {sing}$;

3. (iii) for every $z\in \Sigma ^\mathrm {reg}$, a trivialization $k\xrightarrow \sim z^*\mathcal {L}=k$ given by multiplication by $c_z$.

What is the kernel of the map (1.4.3)? Fix $y\in A$. Then multiplying $a_x$, for $x\in \phi ^{-1}(y)$, and $b_z$, for $z \in \Sigma ^\mathrm {sing}$ such that $g(z)=y$, by a common element of $k^\times$ does not change the descent data (i) or the trivialization (ii), so we obtain the same $(\mathcal {L},\beta )$. The equivalence relation on pairs is realized by the automorphism group $\mathbb {G}_{\mathrm {m}}^C$ of $\phi ^*\mathcal {L}=\mathcal {O}_{\tilde Y}$, which acts on tuples by

\[ (d_j)_{j\in C} \colon ((a_x)_{x\in B},(b_z)_{z\in \Sigma^\mathrm{sing}}, (c_x)_{x\in \Sigma^\mathrm{reg}}) \mapsto ((d_{\psi(x)}a_x), (b_z), (d_{\theta(z)}c_z)). \]

Therefore, $G$ is the torus whose character group is the kernel of the map

(1.4.4)\begin{equation} \mathbb{Z}[B] \oplus \mathbb{Z}[\Sigma^\mathrm{sing}] \oplus \mathbb{Z}[\Sigma^\mathrm{reg}] \to \mathbb{Z}[C] \oplus \mathbb{Z}[A] \end{equation}

with matrix

\[ \begin{bmatrix} \psi & 0 & \theta \\ \phi & \lambda & 0 \end{bmatrix}. \]

The homology complex of $\widetilde{\Gamma} _{Y,\Sigma }$ is

(1.4.5)\begin{equation} \mathbb{Z}[B] \oplus \mathbb{Z}[\Sigma^\mathrm{sing}] \oplus \mathbb{Z}[\Sigma^\mathrm{reg}] \to \mathbb{Z}[C] \oplus \mathbb{Z}[A] \oplus \mathbb{Z} \end{equation}

with differential given by the matrix

\[ \begin{bmatrix} \psi & 0 & \theta \\ -\phi & \lambda & 0 \\ 0 & -\varepsilon & -\varepsilon \end{bmatrix}, \]

where $\varepsilon \colon \mathbb {Z}[\Sigma ^?] \to \mathbb {Z}$ is the augmentation $z\mapsto 1$, for $z\in \Sigma ^?$, $?\in \{\mathrm {reg},\mathrm {sing}\}$. There is an obvious map from the complex (1.4.5) to the complex (1.4.4) which induces an isomorphism on kernels. Since $G$ is by (1.4.2) the maximal multiplicative quotient of $\operatorname {Pic}^{0,\mathrm {lin}}_{(Y,\Sigma )/k}$, this gives the isomorphism (a). The construction is Galois equivariant by transport of structure.

1.5 Functoriality I

Let $g\colon \Sigma \to Y$, $g'\colon \Sigma ' \to Y'$ be generalized moduli on proper curves over $k$ as in the previous section, and suppose we have finite morphisms $f$, $f_\Sigma$ fitting into the following commutative diagram.

(1.5.1)

Then there is an associated pullback morphism

\[ (f,f_\Sigma)^* \colon \operatorname{Pic}_{(Y,\Sigma)/k} \to \operatorname{Pic}_{(Y',\Sigma')/k} \]

taking a pair $(\mathcal {L}, \alpha \colon \mathcal {O}_\Sigma \xrightarrow \sim g^*\mathcal {L})$ to the pair

\[ \big(f^*\mathcal{L},\ f_\Sigma^*\alpha \colon \mathcal{O}_{\Sigma'} \xrightarrow\sim f_\Sigma^*g^*\mathcal{L}=g^{\prime*}(f^*\mathcal{L})\big), \]

which we will simply denote by $f^*$ if no confusion can arise.

To define pushforward, consider the following commutative diagram.

We assume that $f$ is flat, and that $h$ is a closed immersion whose ideal sheaf $\mathcal {I}$ is nilpotent and satisfies

(1.5.2)\begin{equation} \mathcal{N}_{\Sigma\times_YY'/\Sigma}(1+\mathcal{I}) = \{1\}. \end{equation}

We may then define a morphism $f_* =(f,\Sigma )_* \colon \operatorname {Pic}_{(Y',\Sigma ')/k} \to \operatorname {Pic}_{(Y,\Sigma )/k}$ by $f_*\colon (\mathcal {L}',\alpha ') \mapsto (\mathcal {L},\alpha )$, where $\mathcal {L}=\mathcal {N}_f(\mathcal {L}')$, the norm of $\mathcal {L}'$ (see [Reference GrothendieckGro61, 6.5]) and $\alpha$ is given as follows: if $\mathcal {I}=0$ is zero, then (1.5.1) is Cartesian, and $\alpha$ is the composite

\[ \alpha \colon \mathcal{O}_\Sigma \xrightarrow[\mathcal{N}_f(\alpha')]{\sim} \mathcal{N}_f(g^{\prime*}\mathcal{L}') \xrightarrow\sim g^*\mathcal{L} \]

(the second isomorphism given by [Reference GrothendieckGro61, (6.5.8)]). In general, $\alpha '\colon \mathcal {O}_{\Sigma '} \xrightarrow \sim g^{\prime *}\mathcal {L}'$ can at least locally be extended to an isomorphism $\alpha '' \colon \mathcal {O}_{\Sigma \times _YY'} \xrightarrow \sim \mathrm {pr}_2^*\mathcal {L}'$, well-defined up to local sections of $1+\mathcal {I}$. Taking norms, we then get a well-defined global isomorphism $\alpha =\mathcal {N}_{\Sigma \times _Y Y'/\Sigma }(\alpha '')\colon \mathcal {O}_\Sigma \xrightarrow \sim g^*\mathcal {L}$.

The maps $f^*$, $f_*$ preserve $\operatorname {Pic}^0$ in all cases.

Example 1.9 Suppose that $Y$, $Y'$ are smooth over $k$ and absolutely irreducible, and that $\Sigma \subset Y$, $\Sigma '\subset Y'$ are closed subschemes defined by reduced moduli $\mathfrak {m}$, $\mathfrak {m}'$. Let $J_\mathfrak {m}=\operatorname {Pic}^0_{(Y,\Sigma )/k}$, $J'_{\mathfrak {m}'}=\operatorname {Pic}^0_{(Y',\Sigma ')/k}$, be the associated generalized Jacobians. Let $f\colon Y' \to Y$ be a finite morphism with $f^{-1}(\Sigma )^{\mathrm {red}}=\Sigma '$. Then (1.5.2) holds and, therefore, we get morphisms

\[ f^*\colon J_{\mathfrak{m}} \to J'_{\mathfrak{m}'},\quad f_*\colon J'_{\mathfrak{m}'} \to J_{\mathfrak{m}}. \]

If $f'\colon Y' \to Y$ is another finite morphism with $f^{\prime -1}(\Sigma )\supset \Sigma '$, then we get an induced endomorphism $f_*f^{\prime *}\colon J_\mathfrak {m} \to J_\mathfrak {m}$, compatible with the usual correspondence action on $J$ (pullback along $f'$ followed by norm with respect to $f$). For later reference, we will say that the modulus $\mathfrak {m}$ is stable under the correspondence $f_*f^{\prime *}$.

Returning to the general case, assume that $k$ is algebraically closed, that $f$ is flat and that (1.5.2) holds. Write

\[ \mathbb{X} =\operatorname{Hom}(\operatorname{Pic}^{0,\mathrm{lin}}_{(Y,\Sigma)/k},\mathbb{G}_{\mathrm{m}}),\quad \mathbb{X}' =\operatorname{Hom}(\operatorname{Pic}^{0,\mathrm{lin}}_{(Y',\Sigma')/k},\mathbb{G}_{\mathrm{m}}) \]

for the character groups of the linear parts of the generalized Picard schemes. Then $f^*$, $f_*$ induce by functoriality homomorphisms

(1.5.3)\begin{equation} \mathbb{X}(f^*) \colon \mathbb{X}' \to \mathbb{X}, \quad \mathbb{X}(f_*) \colon \mathbb{X} \to \mathbb{X}'. \end{equation}

By Proposition 1.8 and (1.4.4) we have canonical isomorphisms

\begin{align*} \mathbb{X} &=\ker\bigg( \begin{bmatrix} \psi & 0 & \theta \\ \phi & \lambda & 0\end{bmatrix} \colon \mathbb{Z}[B] \oplus \mathbb{Z}[\Sigma^\mathrm{sing}] \oplus \mathbb{Z}[\Sigma^\mathrm{reg}] \to \mathbb{Z}[C] \oplus \mathbb{Z}[A] \bigg)\\ &= H_1(\widetilde\Gamma_{Y,\Sigma},\mathbb{Z}), \end{align*}

where $A=(Y^\mathrm {red})^{\mathrm {sing}}$, $B=\phi ^{-1}(A)$, $C=\pi _0(\widetilde Y)$, and similarly for $\mathbb {X}'$. We now describe the maps (1.5.3) combinatorially, under further hypotheses. Let $A'$, $B'$, $C'$ denote the corresponding sets for $Y'$, and assume the following.

Hypotheses 1.10(ii) and (iii) together imply that (1.5.2) is satisfied. Then $f$ induces maps $A'\to A$, $B'\to B$, $C'\to C$ which we also denote by $f$. The diagram

(1.5.4)

commutes, and so we have the following commutative square.

(1.5.5)

Proof. We first observe that we may assume that Y and $Y'$ are reduced and seminormal. Indeed, the description of the character groups $\mathbb {X}$ and $\mathbb {X}'$ is unchanged after replacing the curves by their seminormalizations. It remains to verify that the induced map on seminormalizations $f^{\mathrm {sn}} \colon Y^{\prime \mathrm {sn}} \to Y^{\mathrm {sn}}$ is flat. But $Y^{\mathrm {sn}}\smallsetminus A = Y^{\mathrm {red}}\smallsetminus A$ is smooth, and so the restriction of $f^{\mathrm {sn}}$ to $Y^{\prime \mathrm {sn}}\smallsetminus A'$ is automatically flat. By hypothesis, there is a neighbourhood $U\subset Y$ of $A$ such that $f \colon U':= f^{-1}(U) \to U$ is étale. Then by [Reference Greco and TraversoGT80, Proposition 5.1], we have a Cartesian square

and, in particular, $f^{\mathrm {sn}}$ restricted to $U^{\prime \mathrm {sn}}$ is étale.

We now compute the dual map $f^* \colon \operatorname {Pic}^{0,\mathrm {lin}}_{(Y,\Sigma )/k} \to \operatorname {Pic}^{0,\mathrm {lin}}_{(Y',\Sigma ')/k}$ which is a morphism of tori (since we are assuming that $Y$ and $Y'$ are seminormal and $\Sigma$, $\Sigma '$ are reduced). As explained in § 1.4, a $k$-point of $\operatorname {Pic}^{0,\mathrm {lin}}_{(Y,\Sigma )/k}$ is represented by a pair $((a_x)_{x\in B},(c_z)_{z\in \Sigma }) \in (k^\times )^B\times (k^\times )^\Sigma$, where $(a_x)$ determines descent data for $\mathcal {O}_{\widetilde Y}$ with respect to the normalization morphism $\phi \colon \widetilde Y \to Y$, and $(c_z)$ determines trivializations $\times c_z\colon k\xrightarrow {\sim }k=z^*\mathcal {O}_{\widetilde Y}$ which descend to a rigidification along $\Sigma$ of the descended line bundle.

Let $a'_{x'}=a_{f(x')}$ ($x'\in B'$) and $c'_{z'}=c_{f(z')}$ ($z'\in \Sigma '$). Then if $(\mathcal {L},\alpha ) \in \operatorname {Pic}^{0,\mathrm {lin}}_{(Y,\Sigma )/k}(k)$ is represented by the pair $((a_x), (c_z))$, the pullback $f^*(\mathcal {L},\alpha )$ is represented by $((a'_{x'}),(c'_{z'}))$. The obvious map

\[ \operatorname{Aut}\mathcal{O}_{\widetilde Y}=(k^\times)^C \to \operatorname{Aut}\mathcal{O}_{\widetilde Y'}=(k^\times)^{C'} \]

is induced by $f\colon C'\to C$ and, therefore, $\mathbb {X}(f^*)$ is induced by the vertical maps $f$ in (1.5.5) as required.

We now compute $\mathbb {X}(f_*)$ assuming Hypotheses 1.10. Let

\[ f^*\colon \begin{cases} \mathbb{Z}[\Sigma] \to \mathbb{Z}[\Sigma'] & \\ \mathbb{Z}[A] \to \mathbb{Z}[A'] & \\ \mathbb{Z}[B] \to \mathbb{Z}[B'] & \end{cases} \]

be the inverse image maps on divisors. By Hypothesis 1.10(i), this means that if $x\in A$ or $x\in B$, then $f^*\colon (x) \mapsto \sum _{f(x')=x}(x')$, and if $z\in \Sigma$, then

\[ f^*\colon (z) \mapsto \sum_{f(z')=z} r_{z'/z}(z') \]

where $r_{z'/z}$ is the ramification degree of $f$ at $z'$. Finally, define $f^*\colon \mathbb {Z}[C] \to \mathbb {Z}[C']$ by

\[ f^*\colon (Z) \mapsto \sum_{f(Z')=Z} [\kappa(Z'):\kappa(Z)]\,(Z'), \]

where $Z\subset \widetilde Y$, $Z'\subset \widetilde Y'$ are connected components. These maps fit into the following diagram.

(1.5.6)

Proof. As in Proposition 1.11, we may assume that $Y$ and $Y'$ are seminormal. Consider again the dual map of tori $f^*\colon \operatorname {Pic}^{0,\mathrm {lin}}_{(Y',\Sigma ')/k} \to \operatorname {Pic}^{0,\mathrm {lin}}_{(Y,\Sigma )/k}$. Let $(\mathcal {L}',\alpha ')\in \operatorname {Pic}^{0,\mathrm {lin}}_{(Y',\Sigma ')/k}(k)$, represented by the pair $((a'_{x'})_{x'\in B'}, (c'_{z'})_{z'\in \Sigma '})$. Since $f$ is étale at $B'$, the normalized map $\tilde f \colon \widetilde Y'\to \widetilde Y$ induces a norm homomorphism

\[ \mathcal{N}_{\tilde f} \colon \Gamma(\phi^{\prime-1}(Y^{\prime\mathrm{sing}}),\mathcal{O}^\times) = (k^\times)^{B'} \to \Gamma(\phi^{-1}(Y^{\mathrm{sing}}),\mathcal{O}^\times) = (k^\times)^{B}, \]

which equals the homomorphism $f_! \colon (k^\times )^{B'} \to (k^\times )^{B}$ given by

\[ f_!\colon (a'_{x'})_{x'\in B'} \mapsto (a_x)_{x\in B},\quad a_x= \prod_{f(x')=x} a'_{x'}. \]

The analogous statement holds for

\[ \mathcal{N}_{f} \colon \Gamma(Y^{\prime\mathrm{sing}},\mathcal{O}^\times) = (k^\times)^{A'} \to \Gamma(Y^{\mathrm{sing}},\mathcal{O}^\times) = (k^\times)^{A}. \]

Since $f$ is étale at $A'$, the square

is Cartesian and, therefore,

\[ \phi^*\circ f_! = f_! \circ \phi^{\prime*} \colon (k^\times)^{B'} \to (k^\times)^A. \]

Next, we consider the rigidification $\alpha '\colon \mathcal {O}_{\Sigma '} \xrightarrow \sim g^{\prime *}\mathcal {L}=g^{\prime *}\mathcal {O}_{\widetilde Y}=\mathcal {O}_{\Sigma '}$ given by multiplication by $(c'_{z'}) \in (k^\times )^{\Sigma '}$. Let $\Sigma '' =f^{-1}(\Sigma )$ be the scheme-theoretic inverse image of $\Sigma$. Then $\Sigma ''=\coprod _{z'\in \Sigma '} \tilde z'$ say, where $\tilde z'\simeq \operatorname {Spec} k[t]/(t^{r_{z'/z}})$. According to the discussion preceding Example 1.9, to compute $f_*(\mathcal {L}',\alpha ')$ we need to extend $\alpha '$ to a rigidification

\[ \alpha'' \colon \mathcal{O}_{\Sigma''} \xrightarrow\sim \mathcal{L}'|_{\Sigma''}=\mathcal{O}_{\Sigma''} \]

and we may as well take $\alpha ''$ to be the sum of the maps $\mathcal {O}_{\tilde z'} \xrightarrow \sim \mathcal {O}_{\tilde z'}$ given by multiplication by $c'_{z'}$. Then $\mathcal {N}(\alpha '')\colon \mathcal {O}_\Sigma \xrightarrow \sim \mathcal {O}_\Sigma$ is multiplication by $(c_z)=\hat f_!(c'_{z'})$, where $\hat f_!\colon (k^\times )^{\Sigma '} \to (k^\times )^\Sigma$ is the map

\[ \hat f_! \colon (c'_{z'}) \mapsto (c_z),\quad c_z=\prod_{f(z')=z} (c'_{z'})^{r_{z'/z}} \]

whose dual is the map $f^*\colon \mathbb {Z}[\Sigma '] \to \mathbb {Z}[\Sigma ]$ defined above.

Finally, we need to compute the action of $\operatorname {Aut} \mathcal {O}_{\widetilde Y'}=(k^\times )^{C'}$. From § 1.4 we know that $d'\in (k^\times )^{C'}$ maps $((a'_{x'})_{x'\in B'}, (c'_{z'})_{z'\in \Sigma '})$ to $((d'_{\psi '(x')}a'_{x'}), (d'_{\theta '(z')}c'_{z'}))$, which under the norm maps to

(1.5.7)\begin{equation} \bigg( \bigg(\prod_{\tilde f(x')=x} d'_{\psi'(x')}a'_{x'}\bigg)_{x\in B}, \bigg(\prod_{\tilde f(z')=z} (d'_{\theta'(z')}c'_{z'})^{r_{z'/z}}\bigg)_{z\in \Sigma} \bigg). \end{equation}

Let $x\in B$ be fixed. Then if $Z=\psi (x)\in C$ is the component containing $x$, and $Z'\in C'$ is a component of $\widetilde Y'$ lying over $Z$, the set $f^{-1}(x)\cap Z'$ has cardinality $[\kappa (Z'):\kappa (Z)]$, since $f$ is étale at $f^{-1}(x)$. Therefore,

\[ \prod_{\tilde f(x')=x} d'_{\psi'(x')} = \prod_{\substack{Z'\in C'\\ f(Z')=\psi(x)}} (d'_{Z'})^{[\kappa(Z'):\kappa(Z)]}. \]

Similarly, let $z\in \Sigma$ be fixed, and $Z=\theta (z)\in C$ the component of $\widetilde Y$ containing it. Then if $Z'\in C'$ is a component of $\widetilde Y'$ lying over $Z$,

\[ \sum_{z'\in f^{-1}(z)\cap Z'} r_{z'/z} = [\kappa(Z'):\kappa(Z)] \]

and, therefore,

\[ \prod_{\tilde f(z')=z} (d'_{\theta'(z')})^{r_{z'/z}} = \prod_{\substack{Z'\in C'\\ f(Z')=\theta(z)}} (d'_{Z'})^{[\kappa(Z'):\kappa(Z)]}. \]

In other words, the pair (1.5.7) equals

\[ (d_{\psi(x)}(f_!a')_x, d_{\theta(Z)}(\hat f_!c')_Z), \]

where

\[ d_Z = \prod_{\substack{Z'\in C'\\ f(Z')=Z}} (d'_{Z'})^{[\kappa(Z'):\kappa(Z)]}. \]

The dual of this map $d'\mapsto d$ is therefore the homomorphism $f^*\colon \mathbb {Z}[C] \to \mathbb {Z}[C']$ defined above.

1.6 Generalized Jacobians over discrete valuation rings

We resume the notations and hypotheses of § 1.3. Let $(x_i)_{i\in I}$ be a nonempty finite family of distinct closed points of $X$, whose residue fields $F_i$ are separable over $F$. Let $\mathfrak {m}=\sum _{i\in I}(x_i)$ be the associated modulus on $X$, and let $J_\mathfrak {m}=\operatorname {Pic}^0_{(X,\mathfrak {m})/F}$ be the generalized Jacobian of $X$ with respect to $\mathfrak {m}$. The semiabelian variety $J_\mathfrak {m}$ is an extension of $J$ by the torus

\[ T_\mathfrak{m}=\bigg(\prod_{i\in I} \mathcal{R}_{F_i/F}\mathbb{G}_{\mathrm{m}}\bigg)/ \mathbb{G}_{\mathrm{m}}. \]

Write $\mathcal {J}_\mathfrak {m}$ for the Néron model of $J_\mathfrak {m}$.

Let $R_i$ be the integral closure of $R$ in $F_i$. Then the inclusion of the points $(x_i)$ in $X$ extends to a unique morphism

\[ \Sigma:= \coprod_{i\in I}\operatorname{Spec} R_i \xrightarrow{g} \mathcal{X}. \]

As $\Gamma (\mathcal {X}_s, \mathcal {O}_{\mathcal {X}_s} ) = k$, the special fibre $g_s \colon \Sigma _s \to \mathcal {X}_s$ is a generalized modulus, in the sense of the previous section. By Proposition 1.4(b) the Néron model $\mathcal {T}_\mathfrak {m}$ of $T_\mathfrak {m}$ equals $(\mathcal {R}_{\Sigma /S} \mathcal {G}_\mathrm {m} )/\mathcal {G}_\mathrm {m}$, and its identity subgroup is $\mathcal {T}_\mathfrak {m}^0 = (\mathcal {R}_{\Sigma /S}\mathbb {G}_{\mathrm {m}} )/\mathbb {G}_{\mathrm {m}}$.

Proof. Let $S'$ be any $S$-scheme. Since $\mathcal {X}/S$ is cohomologically flat and $\Sigma$ is flat over $S$, we have

\begin{gather*} \Gamma(X \times_S S', \mathcal{O}_{\mathcal{X}\times_S S'}) = \Gamma(S', \mathcal{O}_{S'})\quad\text{and}\\ \Gamma(\Sigma \times_S S',\mathcal{O}_{\Sigma\times_S S'}) = \Gamma(\Sigma, \mathcal{O}_\Sigma) \otimes_R \Gamma(S' , \mathcal{O}_{S'}). \end{gather*}

As $\Sigma$ is nonempty, $R \to \Gamma (\Sigma, \mathcal {O}_\Sigma )$ is a split injection of $R$-modules, and therefore $\Gamma (\mathcal {X} \times _S S', \mathcal {O}_{\mathcal {X}\times _S S'}) \to \Gamma (\Sigma \times _S S', \mathcal {O}_{\Sigma \times _S S'})$ is injective.

Let $P_\Sigma$ denote the rigidified Picard functor of [Reference RaynaudRay70, (2.1)]: for any $S$-scheme $S'$, $P_\Sigma (S')$ is the group of equivalence classes of pairs $(\mathcal {L}, \alpha )$, where $\mathcal {L}$ is a line bundle on $\mathcal {X} \times _S S'$, and $\alpha \colon \mathcal {O}_{\Sigma \times _SS'}\xrightarrow \sim (g \times \mathrm {id}_{S'})^* \mathcal {L}$ is a trivialization. Pairs $(\mathcal {L}, \alpha )$ and $(\mathcal {L}', \alpha ')$ are equivalent if there exists an isomorphism $\sigma \colon \mathcal {L} \xrightarrow \sim \mathcal {L}'$ such that $\alpha ' = (g \times \mathrm {id}_{S'})^*(\sigma ) \circ \alpha$. By [Reference RaynaudRay70, (2.3.1–2)], $P_\Sigma$ is a smooth algebraic space in groups over $S$, and we have an exact sequence of algebraic spaces in groups [Reference RaynaudRay70, (2.4.1)]

\[ 0 \to \mathcal{T}_\mathfrak{m}^0 = \mathcal{R}_{\Sigma/S} \mathbb{G}_{\mathrm{m}} /\mathbb{G}_{\mathrm{m}} \to P_\Sigma \xrightarrow{r} P \to 0, \]

where $r$ is the ‘forget the rigidification’ functor. (Since $\mathcal {X}/S$ is cohomologically flat and $f_*\mathcal {O}_\mathcal {X}=\mathcal {O}_S$, one has $\Gamma _X^*=\mathbb {G}_{\mathrm {m}}$.) If $S$ is strictly Henselian, $P_\Sigma$ is a scheme; indeed, $P$ is a scheme, and $\mathcal {T}_\mathfrak {m}^0$ is affine, so by flat descent for affine schemes [Stacks, tag 0245], the $\mathcal {T}_\mathfrak {m}^0$-torsor $P_\Sigma$ over $P$ is representable.

Define the sheaf $P_\mathfrak {m}$ to be the following pushout of fppf sheaves.

(1.6.1)

Explicitly, $P_\mathfrak {m}$ is the sheafification of the functor on $S$-schemes

(1.6.2)\begin{equation} S' \mapsto \mathcal{T}_\mathfrak{m}^0(S') \backslash (P_\Sigma(S')\times \mathcal{T}_\mathfrak{m}(S')), \end{equation}

where $\mathcal {T}_\mathfrak {m}^0(S')$ acts on the product by $a(b, c) = (ab, a^{-1}c)$.

This result implies that $P_\mathfrak {m}$ is determined by its restriction to $(Sm/S)$, the category of essentially smooth $S$-schemes. We can describe this functor explicitly. Let $\mathcal {F}^*$ be the functor on $(Sm/S)$ whose value on $S'$ is the group of equivalence classes of pairs $(\mathcal {L}, \beta = (\beta _i )_{i\in I})$, where $\mathcal {L}$ is a line bundle on $\mathcal {X} \times _S S'$ and for each $i \in I$, $\beta _i \colon \mathcal {O}_{S'} \otimes _R F_i \xrightarrow \sim (x_i \times \mathrm {id}_{S'})^*\mathcal {L}$ is a trivialization of $\mathcal {L}$ at $x_i \times _S S'$. Pairs $(\mathcal {L}, \beta )$ and $(\mathcal {L}', \beta ')$ are equivalent if there exists an isomorphism $\sigma \colon \mathcal {L} \xrightarrow \sim \mathcal {L}'$ and some $u \in \mathcal {O}^\times (S'\otimes _R F)$ such that for every $i$ the following diagram commutes.

(1.6.3)

Note that $u$ is uniquely determined by $\sigma$. If $S' \in (Sm/S)$ is actually an $F$-scheme, then giving a pair $(u, \sigma )$ is the same as giving an isomorphism $(\mathcal {L}, \beta ) \xrightarrow \sim (\mathcal {L}', \beta ')$, since we can absorb $u$ into $\sigma$ and, therefore, the restrictions of $\mathcal {F}^*$ and $P_\Sigma$ to $(Sm/F)$ are equal.

Proof. We have an exact sequence of fppf sheaves

\[ 0 \to \mathbb{G}_{\mathrm{m}} \xrightarrow{\mathrm{diag}} \mathcal{R}_{\Sigma/S} \mathbb{G}_{\mathrm{m}} \times \mathcal{G}_\mathrm{m} \xrightarrow{\psi} P_\Sigma \times \mathcal{R}_{\Sigma/S} \mathcal{G}_\mathrm{m}, \]

where the map $\psi$ on $S'$-valued points is given by

\[ \psi \colon (a, b) \mapsto ((\mathcal{O}_{\mathcal{X}\times_S S'},a\cdot \mathrm{id}_{\mathcal{O}_{\Sigma\times S'}}), a^{-1} b) \in P_\Sigma (S') \times \mathcal{G}_\mathrm{m} (\Sigma \times_S S' ). \]

By definition, $P_\mathfrak {m}$ is the cokernel of $\psi$ in the category of fppf sheaves. As the coimage of $\psi$ is a smooth $S$-group scheme, $P_\mathfrak {m}$ is also the cokernel of $\psi$ in the category of étale sheaves. Let $S' \in (Sm/S)$ and consider the map

\[ \phi_{S'} \colon P_\Sigma (S' ) \times \mathcal{R}_{\Sigma/S} \mathcal{G}_\mathrm{m} (S' ) = P_\Sigma (S' ) \times \mathbb{G}_{\mathrm{m}} (\Sigma_F \times_S S' ) \to \mathcal{F}^* (S' ) \]

given as follows: let $(\mathcal {L}, \alpha )$ represent an element of $P_\Sigma (S')$ and $v \in \mathbb {G}_{\mathrm {m}} (\Sigma _F \times _S S' )$. We map the pair $((\mathcal {L}, \alpha ), v)$ to the equivalence class of $(\mathcal {L}, \beta )$, where $\beta = \alpha \otimes v \colon \mathcal {O}_{\Sigma \times S' \otimes F}\xrightarrow \sim (g \times \mathrm {id}_{S' \otimes F} )^* \mathcal {L}$. It is easy to see that this is well-defined and functorial, and that the resulting sequence of presheaves on $(Sm/S)$

\[ \mathcal{R}_{\Sigma/S}\mathbb{G}_{\mathrm{m}} \times \mathcal{G}_\mathrm{m} \xrightarrow{\psi} P_\Sigma \times \mathcal{R}_{\Sigma/S} \mathcal{G}_\mathrm{m} \xrightarrow{\phi} \mathcal{F}^* \]

is exact. Moreover, for any $(\mathcal {L}, \beta ) \in \mathcal {F}^* (S' )$, there exists a Zariski cover $S'' \to S'$ such that $(\mathcal {L}, \beta )|_{S''}$ is in the image of $\phi _{S''}$. The result follows.

Let $E_\mathfrak {m}$ denote the closure in $P_\mathfrak {m}$ of the zero section. It is contained in

\[ P_\mathfrak{m}' = \ker(\deg \colon P_\mathfrak{m} \to P \to \mathbb{Z}). \]

The analogue of assertion (a) need not hold for $P_\Sigma$: see Example 1.17 after the proof.

Proof. (a) By [Reference RaynaudRay70, (3.3.5)] and Proposition 1.14, $E_\mathfrak {m}$ is an étale algebraic space in groups over $S$. We may therefore compute it by restriction to $(Sm/S)_\mathrm {\acute et}$, using the description of Theorem 1.15, and we may also assume that $S$ is strictly Henselian. In this case, from § 1.3 we have that $E(S)$ is generated by the classes of the line bundles $\mathcal {O}_\mathcal {X} (Y_j )$. Let $\beta _{\mathrm {triv}} = (\beta _{\mathrm {triv},i} )$ be the trivial rigidification of the generic fibre $\mathcal {O}_\mathcal {X} (Y_j )_F = \mathcal {O}_X$ at $(x_i)$. Then $E_\mathfrak {m}$ is generated by the equivalence classes of pairs $(\mathcal {O}_\mathcal {X} (Y_j ),\beta _{\mathrm {triv}})$, and therefore $E_\mathfrak {m} \simeq E$.

(b),(c) We now have an exact sequence

\[ 0 \to \mathcal{T}_\mathfrak{m} \to P_\mathfrak{m}' /E_\mathfrak{m} \to P' /E \to 0 \]

of smooth separated $S$-algebraic spaces in groups, which are therefore separated $S$-group schemes [Reference RaynaudRay70, (3.3.1)], whose generic fibre is the sequence $0 \to T_\mathfrak {m} \to J_\mathfrak {m} \to J \to 0$. As $\mathcal {T}_\mathfrak {m}$ and $P'/E$ are the Néron models of $T$ and $J$, the result follows from Proposition 1.3.

(d) From § 1.1, $\mathcal {T}_{\mathfrak {m},s} /\mathcal {T}_{\mathfrak {m},s}^0 \simeq \operatorname {coker}(e \colon \mathbb {Z} \to \mathbb {Z}^I )$. We then have a commutative diagram of étale sheaves on $(Sm/S)$

whose rows are exact (since $\pi _0$ is right exact). For $S' /S$ smooth, and $(\mathcal {L}, \beta = (\beta _i ))$ representing an element of $\mathcal {F}^* (S' )$, $\beta _i (1)$ is a rational section of $(g_i \times \mathrm {id}_{S'} )^* \mathcal {L}$ so has a well-defined order along the special fibre $\operatorname {ord}_\mathcal {L} \beta _i (1) \in \Gamma (S' , s_* \mathbb {Z})$. If $(\mathcal {L}', \beta ')$ is equivalent to $(\mathcal {L}, \beta )$, then $(\operatorname {ord}_\mathcal {L} \beta _i' (1) - \operatorname {ord}_{\mathcal {L}'} \beta _i (1))_i \in \Gamma (S' , s_* (e\mathbb {Z}))$, which gives a splitting of the bottom row in the diagram (which is, therefore, also exact on the left).

We return to the general case. From § 1.3, $P_s^0 \cap E_s$ is finite constant and cyclic of order $d$, generated by the class of the line bundle $\mathcal {L}'$. Therefore, $P_{\mathfrak {m},s}^0\cap E_{\mathfrak {m},s}$ is finite constant and cyclic of order dividing $d$. Applying the results of § 1.4, we obtain the following.

Finally, we compute the component group $\Phi (J_\mathfrak {m})$.

Theorem 1.19 Suppose that $R$ is strictly Henselian. Then $\Phi (J_\mathfrak {m})$ is canonically isomorphic to the homology of the complex

(1.6.4)\begin{equation} \mathbb{Z}[C] \xrightarrow{(a,h)} \mathbb{Z}^C \oplus \mathbb{Z}^I /e\mathbb{Z} \xrightarrow{b\oplus0} \mathbb{Z}, \end{equation}

where $a$ and $b$ are as in (1.3.2), and $h\colon \mathbb {Z}[C] \to \mathbb {Z}^I/e\mathbb {Z}$ is induced by the map

\begin{align*} C \times I &\to\mathbb{Z} \\ (j, i) &\mapsto h_{ij} := \operatorname{ord}_{\mathcal{O}_\mathcal{X}(Y_j)} \beta_{\mathrm{triv},i} (1). \end{align*}

(Equivalently, $h_{ij}$ is the degree of the divisor $g_i^* Y_j$ on $\operatorname {Spec} R_i$.)

For general $S$ we have $\Sigma \times _S S^{\mathrm {sh}} = \coprod _{\tilde i \in \widetilde I} S_{\tilde i}$, where $S_{\tilde i}$ is the spectrum of a discrete valuation ring finite over $R^{\mathrm {sh}}$. Let $\widetilde C$ be the set of irreducible components of $\mathcal {X} \otimes k^\mathrm {sep}$. Then $\operatorname {Gal}(F^{\mathrm {sep}} /F )$ acts on $\widetilde I$ and $\widetilde C$ through its quotient $\operatorname {Gal}(k^\mathrm {sep}/k)$, and the above gives a $\operatorname {Gal}(k^\mathrm {sep} /k)$-equivariant isomorphism between $\Phi (J_\mathfrak {m})$ and the homology of the complex

(1.6.5)\begin{equation} \mathbb{Z}[\widetilde C] \xrightarrow{(a,h)} \mathbb{Z}^{\widetilde C} \oplus \mathbb{Z}^{\widetilde I} /e\mathbb{Z} \xrightarrow{b\oplus0} \mathbb{Z} \end{equation}

attached to $\mathcal {X} \times _S S^{\mathrm {sh}}$.

In the semistable case we can describe both the character and component groups in terms of the reduced extended graph.

Note that $\theta$ depends only on the labelled graph $(\Gamma _{\mathcal {X}_s,\Sigma _s}, v_0)$. The hypothesis $I=I^\mathrm {reg}$ is satisfied if, for example, $\{x_i\}\subset X(F^{\mathrm {sh}})$.

1.7 Description via Néron models of 1-motives [Reference SuzukiSuz19]

An alternative approach to the determination of the component group $\Phi (J_\mathfrak {m})$ is via duality and the theory of Néron models of $1$-motives developed in [Reference SuzukiSuz19]. We recall some of the notions and results of that paper. Recall that a $1$-motive over $F$ is a two-term complex of group schemes over $F$,

\[ M = [L \xrightarrow{f} G ], \]

where $L$ is étale, free and finitely generated (i.e. $L\otimes _FF^{\mathrm {sep}}\simeq \mathbb {Z}^r$), and $G/F$ is a semiabelian variety. Let $T \subset G$ be its toric part, and $A=G/T$ the abelian variety quotient. We assume here that $L$ and $T$ split over an extension of $F$ in which $R$ is unramified. Then $L$ extends to a local system $\Lambda$ on $S$. Let $\mathcal {G}$ be the Néron model of $G$. By the Néron property, $f$ extends to a morphism $f_S\colon \Lambda \to \mathcal {G}$ of $S$-group schemes and, by definition, the Néron model of $M$ is the complex of $S$-group schemes

\[ \mathcal{M} = [\Lambda \xrightarrow{f_S} \mathcal{G} ]. \]

Its component complex is the complex of $\operatorname {Gal}(k^\mathrm {sep}/k)$-modules

\[ \Phi(M) = [\Lambda_{\bar s} \to \Phi(G)] \]

in degrees $-1$ and $0$. (In [Reference SuzukiSuz19] this complex is denoted by $\mathcal {P}(\mathcal {M})$.)

Let $M'$ be the $1$-motive dual to $M$, so that

\[ M' = [L' \xrightarrow{f'} G' ], \]

where $L'=\operatorname {Hom}(T,\mathbb {G}_{\mathrm {m}})$ is the character group of $T$, and $G'$ is an extension $T' \to G' \to A'$, where $A'$ is the dual abelian variety of $A$, and $T'$ is the torus with character group $L$. Then [Reference SuzukiSuz19, Theorem B] shows that if either:

1. (i) $A$ has semistable reduction; or

2. (ii) $k$ is perfect;

there is a canonical isomorphism, in the derived category of $\operatorname {Gal}(k^\mathrm {sep}/k)$-modules, between $\Phi (M')$ and $\mathrm {RHom}(\Phi (M),\mathbb {Z})[1]$.

Now let $\mathcal {X}/S$ be as in § 1.6. We will assume that $R$ is strictly Henselian. Suppose that all $n_j$ are zero (which holds, for example, if $k$ is perfect), that $\delta =1$, and that the points $(x_i)_{i\in I}$ are all $F$-rational.

Since $J$ is autodual, the dual $1$-motive to $J_\mathfrak {m}$ is the $1$-motive

\[ M = [\mathbb{Z}[I]_0 \to J],\quad i \mapsto \mathcal{O}_X(x_i) \]

whose component complex $\Phi (M)$ is the complex $[\mathbb {Z}[I]_0 \to \Phi (J) ]$ of abelian groups, concentrated in degrees $-1$ and $0$. Using the description (1.3.1) of $\Phi (J)$, this is isomorphic to the complex $[\mathbb {Z}[I]_0 \to \mathbb {Z}^{C,0}/a(\mathbb {Z}^C) ]$.

As $\delta =1$, by [Reference RaynaudRay70, (8.1.2)] the complex (1.3.1) has only one nonzero homology group, namely $\ker (b)/\operatorname {im}(a)=\Phi (J)$, and the map $a$ is given by the intersection pairing on the components of the special fibre. Therefore $\Phi (M)$ is quasi-isomorphic to the following complex.

(1.7.1)

Here ${}^th\colon \mathbb {Z}[I]_0 \to \mathbb {Z}^C$ is the transpose of $h$, and the term $\mathbb {Z}^C$ is in degree $0$. The dual of (

) is the following complex.

The assumption $n_j=0$ ensures that $a$ is symmetric, and that $i$ and $b$ are transposes of one another, by (1.3.2). Assuming that one of assumptions (i) or (ii) above holds, we then recover the description of $\Phi (J_\mathfrak {m})$ as the homology of (1.6.4).

1.8 Functoriality II

We will need to understand the action of correspondences on generalized Jacobians and their Néron models.

Suppose that we have two smooth geometrically connected curves $X$, $X'$ over $F$, with regular models $\mathcal {X}$, $\mathcal {X}'$ satisfying the hypotheses of § 1.3. Let $\mathfrak {m}=\sum _{i\in I}(x_i)$, $\mathfrak {m}'=\sum _{j\in I'}(x'_j)$, be nonzero moduli on $X$, $X'$. As in § 1.6 we assume that the points $x_i$, $x'_j$ are distinct, and that their residue fields

\[ F_i=\kappa(x_i),\quad F'_j = \kappa(x'_j) \]

are separable over $F$. Let $R_i$, $R'_j$ denote the integral closures of $R$ in $F_i$, $F'_j$, and $\Sigma =\coprod _{i\in I}\operatorname {Spec} R_i$, $\Sigma '=\coprod _{j\in I'}\operatorname {Spec} R_j'$. Let $J_\mathfrak {m}$, $J'_{\mathfrak {m}'}$ be the associated generalized Jacobians.

Let $f\colon X' \to X$ be a finite morphism such that $f^{-1}(\Sigma _F)=\Sigma '_F$ as sets. We write $f\colon I' \to I$ also for the induced surjective map on index sets. For $j\in I'$, denote by $r_j$ the ramification degree of $f$ at $x'_j$.

The discussion in § 1.5 applies, and since $f^*$, $f_*$ preserve line bundles of degree zero, we obtain morphisms

\begin{align*} f^* \colon J_\mathfrak{m} \to J'_{\mathfrak{m}'}, \quad f_* \colon J'_{\mathfrak{m}'} \to J_\mathfrak{m}. \end{align*}

By the universal Néron property, they extend uniquely to morphisms $f^*$, $f_*$ of the Néron models $\mathcal {J}_\mathfrak {m}$, $\mathcal {J}'_{\mathfrak {m}'}$. Let the induced homomorphisms of character groups be $\mathbb {X}(f^*)$, $\mathbb {X}(f_*)$ and of component groups $\Phi (f^*)$, $\Phi (f_*)$. In the next section we will need to know explicitly the restriction of these maps to the tori $\mathcal {T}_\mathfrak {m}$, $\mathcal {T}'_{\mathfrak {m}'}$. For parts (a) and (b) below, recall that we have a canonical isomorphism

\[ \operatorname{Hom}(T_{\mathfrak{m}}\otimes F^{\mathrm{sep}},\mathbb{G}_{\mathrm{m}}) \xrightarrow\sim \mathbb{Z}[\Sigma(F^{\mathrm{sep}})]^{\deg=0} \]

and similarly for $T'_{\mathfrak {m}'}$.