Equicharacteristic etale cohomology in dimension one

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.