Models of torsors and the fundamental group scheme

Abstract

Given a relative faithfully at pointed scheme over the spectrum of a discrete valuation ring X !S, this paper is motivated by the study of the natural morphism from the fundamental group scheme of the generic ber X to the generic ber of the fundamental group scheme of X. Given a torsor T !X under an a ne group scheme G over the generic ber of X, we address the question of nding a model of this torsor over X, focusing in particular on the case where G is nite. We provide several answers to this question, showing for instance that, when X is integral and regular of relative dimension 1, such a model exists on some model X0 of X obtained by performing a nite number of N eron blowups along a closed subset of the special ber of X. Furthermore, we show that when G is etale, then we can nd a model of T !X under the action of some smooth group scheme. In the rst part of the paper, we show that the relative fundamental group scheme of X has an interpretation as the Tannaka Galois group of a Tannakian category constructed starting from the universal torsor.