#### Vol. 4, No. 1, 2010

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An Euler–Poincaré bound for equicharacteristic étale sheaves

### Carl A. Miller

Vol. 4 (2010), No. 1, 21–45
##### 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 this 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 different types of sheaves on a characteristic-$p$ scheme. In the paper we prove a one-dimensional version of this correspondence, considering both local and global settings.

##### Keywords
characteristic-$p$ curves, Grothendieck–Ogg–Shafarevich formula, étale sheaves, Riemann–Hilbert correspondence, Frobenius endomorphism, minimal roots
##### Mathematical Subject Classification 2000
Primary: 14F20
Secondary: 13A35, 14F30