Equicharacteristic etale cohomology in dimension one

2009/06/22 by Miller, Carl A.
#14F20 (Primary) 13A35 #14F30 (Secondary) #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.0906.4093

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.

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