Ramification and cleanliness

Abstract : This article is devoted to studying the ramification of Galois torsors and of $\ell$-adic sheaves in characteristic $p>0$ (with $\ell\not=p$). Let $k$ be a perfect field of characteristic $p>0$, $X$ be a smooth, separated and quasi-compact $k$-scheme, $D$ be a simple normal crossing divisor on $X$, $U=X-D$, $\Lambda$ be a finite local ${\mathbb Z}_\ell$-algebra, $F$ be a locally constant constructible sheaf of $\Lambda$-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 one 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.).
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Submitted on : Friday, June 17, 2011 - 5:02:23 PM
Last modification on : Thursday, November 15, 2018 - 11:56:11 AM

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  • HAL Id : hal-00601448, version 1
  • ARXIV : 1007.3873

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Ahmed Abbes, Takeshi Saito. Ramification and cleanliness. Masanari Kida; Noriyuki Suwa ; Shinichi Kobayashi. RIMS Workshop, Dec 2010, Kyoto, Japan. Kyoto University, Algebraic number theory and related topics 2010, B32, pp.19-29, 2012, Reserach Institute for Mathematical Sciences Kôkyûroku bessatsu. 〈hal-00601448〉

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