Effective models and extension of torsors over a discrete valuation ring of unequal characteristic

Abstract

Let R be a discrete valuation ring of unequal characteristic with fraction field K which contains a primitive p^2-th root of unity. Let X be a faithfully flat R-scheme and G be a finite abstract group. Let us consider a G-torsor Y_K\to X_K and let Y be the normalization of X_K in Y. If G=Z/p^n Z, n<3, under some hypothesis on X, we attach some invariants to Y_K \to X_K. If p>2, we determine, through these invariants, when Y\to X has a structure of torsor which extends that of Y_K\to X_K. Moreover we explicitly calculate the effective model (defined by Romagny) of the action of G on Y.