On the abelian fundamental group scheme of a family of varieties

Let S be a connected Dedekind scheme and X an S -scheme provided with a section x . We prove that the morphism between fundamental group schemes π 1 ( X, x ) ab → π 1 ( Alb X/S , ) induced by the canonical morphism from X to its Albanese scheme Alb X/S (when the latter exists) fits in an exact sequence of group schemes 0 → ( NS X/S τ ) ⋎ → π 1 ( X, x ) ab → π 1 ( Alb X/S , ) → 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 η of X can be extended to a finite and commutative pointed torsor over X .