Comparison between the fundamental group scheme of a relative scheme and that of its generic fiber

TL;DR

This paper investigates the relationship between the fundamental group scheme of a scheme's generic fiber and the generic fiber of its fundamental group scheme, establishing faithful flatness and conditions for extension of torsors.

Contribution

It proves the natural morphism between these fundamental group schemes is always faithfully flat and characterizes when torsors extend from the generic fiber to the entire scheme.

Findings

The morphism is always faithfully flat.

Provides criteria for extending torsors over the generic fiber.

Examples where the morphism is an isomorphism.

Abstract

We show that the natural morphism $\phi:\pi_1(X_{\eta},x_{\eta})\to \pi_1(X,x)_{\eta}$ between the fundamental group scheme of the generic fiber $X_{\eta}$ of a scheme $X$ over a connected Dedekind scheme and the generic fiber of the fundamental group scheme of $X$ is always faithfully flat. As an application we give a necessary and sufficient condition for a finite, dominated pointed $G$-torsor over $X_{\eta}$ to be extended over $X$. We finally provide examples where $\phi:\pi_1(X_{\eta},x_{\eta})\to \pi_1(X,x)_{\eta}$ is an isomorphism..