Models of torsors and the fundamental group scheme

Given a relative faithfully flat 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 fiber Xη to the generic fiber of the fundamental group scheme of X. Given a torsor T → Xη under an affine group scheme G over the generic fiber of X, we address the question to find a model of this torsor over X, focusing in particular on the case where G is finite. We obtain partial 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 of Xη obtained by performing a finite number of Néron blow-ups along a closed subset of the special fiber of X. In the first part 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