Let $R$ be a discrete valuation ring with fraction field $K$. Let $X$ be a proper and faithfully flat $R$-scheme, endowed with a section $x \in X(R)$, with connected and reduced generic fibre $X_{ \eta}$. Let $f: Y \rightarrow X_{ \eta}$ be a finite Nori-reduced $G$-torsor. In this paper we provide a useful criterion to extend $f: Y \rightarrow X_{ \eta}$ to a torsor over $X$. Furthermore in the particular situation where $R$ is a complete discrete valuation ring of residue characteristic $p>0$ and $X\to \text{ Spec}(R)$ is smooth we apply our criterion to prove that the natural morphism $\psi^{ (p')}: \pi(X_{ \eta},x_{ \eta})^{ (p')}\to \pi(X,x)_{ \eta}^{ (p')}$ between the prime-to-$p$ fundamental group scheme of $X_{ \eta}$ and the generic fibre of the prime-to-$p$ fundamental group scheme of $X$ is an isomorphism. This generalizes a well known result for the \'etale fundamental group. The methods used are purely tannakian.
Categories: math.AG