RAMIFICATION AND CLEANLINESS

摘要

This article is devoted to studying the ramification of Galois torsors and of ?-adic sheaves in characteristic p ? 0 (with ? ‰ 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, Λ a finite local Z_?-algebra and F a locally constant constructible sheaf of Λ-modules on U. We introduce a boundedness condition on the ramification of 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 F. The cleanliness condition extends the one introduced by Kato for rank 1 sheaves. Roughly speaking, it means that the ramification of 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 F. Some cases of this formula have been previously proved by Kato and by the second author (T. S.).