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∈X(R), with connected and reduced generic fibre Xη. Let f:Y→Xη be a finite Nori-reduced G-torsor. In this paper we provide a useful criterion to extend f:Y→Xη 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→Spec(R) is smooth we apply our criterion to prove that the natural morphism ψ(p′):π(Xη,xη)(p′)→π(X,x)(p′)η between the prime-to-p fundamental group scheme of Xη 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.