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Ramification theory for varieties over a local field

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Abstract

We define generalizations of classical invariants of wild ramification for coverings on a variety of arbitrary dimension over a local field. For an ℓ-adic sheaf, we define its Swan class as a 0-cycle class supported on the wild ramification locus. We prove a formula of Riemann-Roch type for the Swan conductor of cohomology together with its relative version, assuming that the local field is of mixed characteristic.

We also prove the integrality of the Swan class for curves over a local field as a generalization of the Hasse-Arf theorem. We derive a proof of a conjecture of Serre on the Artin character for a group action with an isolated fixed point on a regular local ring, assuming the dimension is 2.

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Authors and Affiliations

  1. Department of Mathematics, University of Chicago, Chicago, IL, 60637, USA

    Kazuya Kato

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

    Takeshi Saito

Authors
  1. Kazuya Kato
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  2. Takeshi Saito
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Correspondence to Takeshi Saito.

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Kato, K., Saito, T. Ramification theory for varieties over a local field. Publ.math.IHES 117, 1–178 (2013). https://doi.org/10.1007/s10240-013-0048-z

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  • DOI: https://doi.org/10.1007/s10240-013-0048-z

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