Given a relative faithfully flat pointed scheme over the spectrum of a discrete valuation ring $X\rightarrow S$, this paper is motivated by the study of the natural morphism from the fundamental group scheme of the generic fiber $X_{\unicode[STIX]{x1D702}}$ to the generic fiber of the fundamental group scheme of $X$. Given a torsor $T\rightarrow X_{\unicode[STIX]{x1D702}}$ under an affine group scheme $G$ over the generic fiber of $X$, we address the question of finding a model of this torsor over $X$, focusing in particular on the case where $G$ is finite. 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 $X^{\prime }$ of $X_{\unicode[STIX]{x1D702}}$ obtained by performing a finite number of Néron blowups along a closed subset of the special fiber of $X$. Furthermore, we show that when $G$ is étale, then we can find a model of $T\rightarrow X_{\unicode[STIX]{x1D702}}$ under the action of some smooth group scheme. In the first 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.
Failure of Krull-Schmidt for invertible lattices over a discrete valuation ring
