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The characteristic class and ramification of an ℓ-adic étale sheaf

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Abstract

We introduce the characteristic class of an ℓ-adic étale sheaf using a cohomological pairing due to Verdier (SGA5). As a consequence of the Lefschetz–Verdier trace formula, its trace computes the Euler–Poincaré characteristic of the sheaf. We compare the characteristic class to two other invariants arising from ramification theory. One is the Swan class of Kato-Saito [17] and the other is the 0-cycle class defined by Kato for rank 1 sheaves in [16].

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Authors and Affiliations

  1. CNRS, LAGA, Université de Paris 13, 99, Av. J.-B. Clément, 93430, Villetaneuse, France

    Ahmed Abbes

  2. Department of Mathematical Sciences, University of Tokyo, Tokyo, 153-8914, Japan

    Takeshi Saito

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  1. Ahmed Abbes
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  2. Takeshi Saito
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Correspondence to Takeshi Saito.

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Dedicated to Luc Illusie, with admiration

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Abbes, A., Saito, T. The characteristic class and ramification of an ℓ-adic étale sheaf. Invent. math. 168, 567–612 (2007). https://doi.org/10.1007/s00222-007-0040-7

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  • DOI: https://doi.org/10.1007/s00222-007-0040-7

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