AN EULER-POINCARE ́ BOUND FOR EQUICHARACTERISTIC ÉTALE SHEAVES

Abstract. The Grothendieck-Ogg-Shafarevich formula expresses the Euler characteristic of an étale sheaf on a characteristic-p curve in terms of local data. The purpose of this paper is to prove an equicharacteristic version of the G-O-S formula (a bound, rather than an equality). This follows work of R. Pink. The basis for the proof of this result is the characteristic-p “Riemann-Hilbert ” correspondence, which is a functorial relationship between two dif-ferent types of sheaves on a characteristic-p scheme. In the paper we prove a one-dimensional version of this Riemann-Hilbert correspondence, considering both local and global settings. 1