Crew's Euler characteristic formula fails for nonzero slopes

Abstract

A result of Crew implies that for an etale Galois p-cover of smooth proper varieties over a field of characteristic p, the alternating sum of the p-ranks of the cohomology groups behave like Euler characteristics in characteristic 0. That is, the sum for the top curve equals the sum for the bottom curve times the degree of the cover. We exhibit an example to show that this relation fails in general if the p-rank is replaced by the part of some fixed slope (other than 0).