## Tohoku Mathematical Journal 2011 December SECOND SERIES VOL. 63, NO. 4

 Tohoku Math. J. 63 (2011), 775-853

Title RAMIFICATION AND CLEANLINESS

Author Ahmed Abbes and Takeshi Saito

(Received July 22, 2010, revised July 14, 2011)
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 {\em 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 {\em 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.).

2000 Mathematics Subject Classification. Primary 14F20; Secondary 11S15, 57R20.

Key words and phrases. $\ell$-adic sheaves, clean sheaves, wild ramification, characteristic cycle.