On the degeneration of étale $\Bbb Z/p\Bbb Z$ and $\Bbb Z/p^2\Bbb Z$-torsors in equal characteristic $p>0$

Abstract

Let $R$ be a complete discrete valuation ring of equal characteristic $p>0$. In this paper we investigate finite and flat morphisms $f:Y\to X$ between formal $R$-schemes which have the structure of an étale $\Bbb Z/p^n\Bbb Z$-torsor above the generic fiber of $X$, for $n=1,2$, with some extra geometric conditions on $X$ and $Y$. In the case $n=1$, we prove that $f$ has the structure of a torsor under a finite and flat $R$-group scheme of rank $p$ and we describe the group schemes that arise as the group of the torsor. In the case $n=2$, we describe explicitly how the Artin-Schreier-Witt equations describing $f$ on the generic fiber, locally, degenerate. Moreover, in some cases where $f$ has the structure of a torsor under a finite and flat $R$-group scheme of rank $p^2$, we describe the group schemes of rank $p^2$ which arise in this way.