Equicharacteristic etale cohomology in dimension one

TL;DR

This paper extends the Grothendieck-Ogg-Shafarevich formula to equicharacteristic sheaves on curves, using a characteristic-p Riemann-Hilbert correspondence and introducing a new invariant called the minimal root index.

Contribution

It proves a bound version of the G-O-S formula for equicharacteristic sheaves and establishes a local-global correspondence via O_{F, X}-modules.

Findings

Established a version of the G-O-S formula for equicharacteristic sheaves.

Defined the minimal root index as a measure of local complexity.

Proved a Riemann-Hilbert correspondence for curves in characteristic p.

Abstract

The Grothendieck-Ogg-Shafarevich formula expresses the Euler characteristic of an etale sheaf on a curve in terms of local data. The purpose of this paper is to prove a version of the G-O-S formula which applies to equicharacteristic sheaves (a bound, rather than an equality). This follows a proposal of R. Pink. The basis for the result is the characteristic-p "Riemann-Hilbert" correspondence, which relates equicharacteristic etale sheaves to O_{F, X}-modules. In the paper we prove a version of this correspondence for curves, considering both local and global settings. In the process we define an invariant, the "minimal root index," which measures the local complexity of an O_{F, X}-module. This invariant provides the local terms for the main result.