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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DOI: 10.1007/s10240-013-0048-z
Kazuya Kato 1; Takeshi Saito 2
@article{PMIHES_2013__117__1_0,
author = {Kazuya Kato and Takeshi Saito},
title = {Ramification theory for varieties over a local field},
journal = {Publications Math\'ematiques de l'IH\'ES},
pages = {1--178},
year = {2013},
publisher = {Springer-Verlag},
volume = {117},
doi = {10.1007/s10240-013-0048-z},
zbl = {1290.14011},
language = {en},
url = {https://pmihes.centre-mersenne.org/articles/10.1007/s10240-013-0048-z/}
}
TY - JOUR AU - Kazuya Kato AU - Takeshi Saito TI - Ramification theory for varieties over a local field JO - Publications Mathématiques de l'IHÉS PY - 2013 SP - 1 EP - 178 VL - 117 PB - Springer-Verlag UR - https://pmihes.centre-mersenne.org/articles/10.1007/s10240-013-0048-z/ DO - 10.1007/s10240-013-0048-z LA - en ID - PMIHES_2013__117__1_0 ER -
%0 Journal Article %A Kazuya Kato %A Takeshi Saito %T Ramification theory for varieties over a local field %J Publications Mathématiques de l'IHÉS %D 2013 %P 1-178 %V 117 %I Springer-Verlag %U https://pmihes.centre-mersenne.org/articles/10.1007/s10240-013-0048-z/ %R 10.1007/s10240-013-0048-z %G en %F PMIHES_2013__117__1_0
Kazuya Kato; Takeshi Saito. Ramification theory for varieties over a local field. Publications Mathématiques de l'IHÉS, Volume 117 (2013), pp. 1-178. doi: 10.1007/s10240-013-0048-z
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