Abstract
This article is devoted to studying the ramification of Galois torsors and of $\ell $-adic sheaves in characteristic $p>0$ (with $\ell \ne p$). Let $k$ be a perfect field of characteristic $p>0$, $X$ a smooth, separated and quasi-compact $k$-scheme, $D$ a simple normal crossing divisor on $X$, $U=X-D$, $\Lambda$ a finite local $\mathbb{Z}_{\ell} $-algebra and $\mathscr{F}$ a locally constant constructible sheaf of $\Lambda$-modules on $U$. We introduce a boundedness condition on the ramification of $\mathscr{F}$ along $D$, and study its main properties, in particular, some specialization properties that lead to the fundamental notion of cleanliness and to the definition of the characteristic cycle of $\mathscr{F}$. The cleanliness condition extends the one introduced by Kato for rank 1 sheaves. Roughly speaking, it means that the ramification of $\mathscr{F}$ along $D$ is controlled by its ramification at the generic points of $D$. Under this condition, we propose a conjectural Riemann-Roch type formula for $\mathscr{F}$. Some cases of this formula have been previously proved by Kato and by the second author (T. S.).
