| Résume | Let $R$ be a discrete valuation ring of unequalcharacteristic with fraction field $K$ which contains a primitive$p^2$-{th} root of unity. Let $X$ be a faithfully flat $R$-scheme and let $G$ be a finite abstract group. Let us consider a $G$-torsor $Y_K\rightarrow X_K$ and let $Y$ be the normalization of $X$ in $Y_K$. If $G=\mathbb{Z}/p^n$ with $n\le 2$, under some hypothesis on $X$, we attach some invariants to $Y_K\rightarrow X_K$. If $p>2$, we determine through these invariants when $Y\rightarrow X$ has a structure of a $G$-torsor extending the torsor $Y_K\rightarrow X_K$. Moreover we explicitly calculate the effective model (defined by Romagny) of the action of $G$ on $Y$. The explicit classification of $R$-group schemes isomorphic to $\mathbb{Z}/p^n$ ($n\le 2$) over $K$ plays a crucial role. For $n=1$ this classification was already known and the problem of extension of $\mathbb{Z}/p$-torsors has already been studied (for instance by Raynaud, Green-Matignon, Henrio, Sa\"idi) when $X$ is a formal curve. The problem of extension of torsors arises naturally in the local study of group actions on an $R$-scheme. " |
