On localizations of the characteristic classes of l-adic sheaves and conductor formula in characteristic p>0

TL;DR

This paper extends the Grothendieck-Ogg-Shafarevich formula to higher dimensions using characteristic classes of l-adic sheaves, providing a localized version and a conductor formula in characteristic p>0.

Contribution

It introduces a localized characteristic class formula for l-adic sheaves on higher-dimensional schemes, generalizing previous results and assuming strong resolution of singularities.

Findings

Proved a localized characteristic class formula for l-adic sheaves.

Established a conductor formula in equal characteristic.

Extended the Grothendieck-Ogg-Shafarevich formula to higher dimensions.

Abstract

The Grothendieck-Ogg-Shafarevich formula calculates the l-adic Euler-Poincare number of an l-adic sheaf on a curve by an invariant produced by the wild ramification of the l-adic sheaf named Swan class. A. Abbes, K. Kato and T. Saito generalize this formula to any dimensional scheme. In this paper, assuming the strong resolution of singularities we prove a localized version of a formula proved by A. Abbes and T. Saito using the characteristic class of an l-adic sheaf. As an application, we prove a conductor formula in equal characteristic.