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HOME > Table of Contents and Abstracts > Vol. 63, No. 4
Tohoku Mathematical Journal
2011
December
SECOND SERIES VOL. 63, NO. 4
Tohoku Math. J.
63 (2011), 775-853
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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.
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