On the ramification of non-abelian Galois coverings of degree $p^3$

Abstract

The refined Swan conductor is defined by K.\ Kato \cite{KK2}, and generalized by T.\ Saito \cite{wild}. In this part, we consider some smooth $l$-adic \'{e}tale sheaves of rank $p$ such that we can be define the $rsw$ following T.\ Saito, on some smooth dense open subscheme $U$ of a smooth separated scheme X of finite type over a perfect fields $\kappa$ of characteristic $p>0$. We give an explicit expression of $rsw(\mathcal{F})$ in some situation. As a consequence, we show that it is integral.