# Ramification theory for varieties over a perfect field

### Abstract

For an $\ell$-adic sheaf on a variety of arbitrary dimension over a perfect field, we define the Swan class measuring the wild ramification as a $0$-cycle class supported on the ramification locus. We prove a Lefschetz trace formula for open varieties and a generalization of the Grothendieck-Ogg-Shararevich formula using the Swan class.

## Authors

Kazuya Kato

Department of Mathematics
Kyoto University
606-8502 Kyoto
Japan

Takeshi Saito

Department of Mathematical Sciences
University of Tokyo
153-8914 Tokyo
Japan