Tohoku Mathematical Journal

Ramification and cleanliness

Ahmed Abbes and Takeshi Saito

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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.).

Article information

Source
Tohoku Math. J. (2), Volume 63, Number 4 (2011), 775-853.

Dates
First available in Project Euclid: 6 January 2012

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1325886290

Digital Object Identifier
doi:10.2748/tmj/1325886290

Mathematical Reviews number (MathSciNet)
MR2872965

Zentralblatt MATH identifier
1239.14012

Subjects
Primary: 14F20: Étale and other Grothendieck topologies and (co)homologies
Secondary: 11S15: Ramification and extension theory 57R20: Characteristic classes and numbers

Keywords
$\ell$-adic sheaves clean sheaves wild ramification characteristic cycle

Citation

Abbes, Ahmed; Saito, Takeshi. Ramification and cleanliness. Tohoku Math. J. (2) 63 (2011), no. 4, 775--853. doi:10.2748/tmj/1325886290. https://projecteuclid.org/euclid.tmj/1325886290


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