Ramification theory for varieties over a local field

TL;DR

This paper introduces generalized invariants for wild ramification on varieties over local fields, including a Swan class and a Riemann-Roch type formula, extending classical results to higher dimensions.

Contribution

It defines a Swan class for l-adic sheaves on varieties of arbitrary dimension and proves a Riemann-Roch type formula and integrality results, extending classical ramification theory.

Findings

Defined Swan class as a 0-cycle supported on wild ramification locus.

Proved a Riemann-Roch type formula for Swan conductors in mixed characteristic.

Established the integrality of the Swan class for curves, generalizing Hasse-Arf theorem.

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.