Extension of torsors and prime to \(p\) fundamental group scheme (Q2152474)

Suppose \(X\) is a finite type scheme, faithfully flat over a complete discrete valuation ring \(R\) with field of fraction \(K\). Let \(X_K\) denote its generic fibre and let \(f: Y \to X_K\) be a \(G\)-torsor, with \(G\) an affine, finite, and algebraic \(K\)-group scheme. In this paper, the authors provide a criterion to extend \(f\) to a torsor over \(X\) whose generic fibre is isomorphic to \(f\). More specifically, they prove the following theorem. Theorem. Let \(R\) be a complete DVR. Suppose there exist a finite field extension \(K'/K\) and a finite and faithfully flat morphism \(\varphi: Z \to X\) such that \(\varphi^*\varphi_*\mathcal{O}_Z\) is a free \(\mathcal{O}_Z\)-module and \(\varphi_K = \lambda \circ f_{K'}\) where \(\lambda: X_{K'} \to X_K\) and \(F_{K'} : Y_{K'} \to X_{K'}\) are the natural pullback morphisms. If moreover \(\mathcal{O}_Z(Z) = R'\) where \(R'\) is the integral closure of \(R\) in \(K'\), then there is a \(M\)-torsor \(f_1: Y_1 \to X\), for some quasi-finite affine and flat group scheme \(M\) over \(R\), whose generic fibre equals the \(G\)-torsor \(f:Y \to X_K\). As an application of the main theorem, the authors provide a result to the base change behavior of the fundamental group scheme. Corollary. Let \(R\) be a (not necessarily complete) DVR and \(X\), \(Y\) as in the last theorem. Assume \(Y \to X_K\) can be extended to some finite and faithfully flat morphism \(\varphi: Z \to X\) such that \(\varphi^*\varphi_*\mathcal{O}_Z\) is a free \(\mathcal{O}_Z\)-module. Then there exists a \(M\)-torsor \(f_1: Y_1 \to X\) for some quasi-finite affine and flat group scheme \(M\) over \(R\), whose generic fibre equals the \(G\)-torsor \(f: Y \to X_K\).