Towers of torsors over a field

Abstract

Let $X$ be a projective, connected and smooth scheme defined over an algebraically closed field $k$. In this paper we prove that a tower of finite torsors (i.e., under the action of finite $k$-group schemes) can be dominated by a single finite torsor. Let $G$ be any finite $k$-group scheme and $Y$ any $G$--torsor over $X$ pointed in $y\,\in\, Y(k)$; we define over $Y$, which may not be reduced, in a very natural way the categories of Nori-semistable and essentially finite vector bundles. These categories are proved to be Tannakian. Their Galois $k$-group schemes $\pi^S(Y,\,y)$ and $\pi^N(Y,\,y)$, respectively, thus generalize the $S$--fundamental and the Nori fundamental group schemes. The latter still classifies all the finite torsors over $Y$, pointed over $y$. We also prove that they fit in short exact sequences involving $\pi^S(X,\,x)$ and $\pi^N(X,\, x)$ respectively, where $x$ is the image of $y$.