Extension of torsors and prime to $p$ fundamental group scheme

Abstract

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 $f:Y\to X_{\eta }$ be a finite Nori-reduced $G$-torsor. In this paper we provide a useful criterion to extend $f:Y\to X_{\eta }$ 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 ${\psi }^{(p^{\prime })}:\pi (X_{\eta },x_{\eta }{)}^{(p^{\prime })}\to \pi (X,x{)}_{\eta }^{(p^{\prime })}$ 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 étale fundamental group. The methods used are purely tannakian.

To appear in Annals de l'Institut Fourier

References in corpus (1)

• Finite torsors on projective schemes defined over a discrete valuation ring