Let $S$ be a connected Dedekind scheme and $X$ an $S$-scheme provided with a section $x$. We prove that the morphism of fundamental group schemes $\pi_1(X,x)^{ ab}\to \pi_1(\mathbf{ Alb}_{ X/S},0_{ \mathbf{ Alb}_{ X/S}})$ induced by the canonical morphism from $X$ to its Albanese scheme $\mathbf{ Alb}_{ X/S}$ (when the latter exists) fits in an exact sequence of group schemes $0\to (\mathbf{ NS}^{ \tau}_{ X/S})^{ \vee}\to \pi_1(X,x)^{ ab}\to \pi_1(\mathbf{ Alb}_{ X/S},0_{ \mathbf{ Alb}_{ X/S}}) \to 0$ where the kernel is a finite and flat $S$-group scheme. Furthermore we prove that any finite and commutative quotient pointed torsor over the generic fiber $X_{ \eta}$ of $X$ can be extended to a finite and commutative pointed torsor over $X$.
Categories: math.AG math.NT