On the Abelian Fundamental Group Scheme of a Family of Varities

Abstract

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 $0\to ({\mathbf{NS}}_{X/S}^{\tau }{)}^{\vee }\to {\pi }_1(X,x{)}^{ab}\to {\pi }_1({\mathbf{Alb}}_{X/S},0_{{\mathbf{Alb}}_{X/S}})\to 0$ 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 $X$.

arXiv comment18 pages, minor changes, final version