An Euler-Poincaré bound for equicharacteristic étale sheaves. (English) Zbl 1185.14016

Miller, Carl A.

Algebra Number Theory 4, No. 1, 21-45 (2010).

Summary: 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.

MSC:

14F20	Étale and other Grothendieck topologies and (co)homologies
13A35	Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure
14F30	\(p\)-adic cohomology, crystalline cohomology

Keywords:

characteristic-\(p\) curves; Grothendieck-Ogg-Shafarevich formula; étale sheaves; Riemann-Hilbert correspondence; Frobenius endomorphism; minimal roots