The étale cohomology of $p$-torsion sheaves. I

Abstract:

This paper generalizes a formula of Grothendieck, Ogg, and Shafarevich that expresses the Euler-Poincaré characteristic of a constructible sheaf of ${F_l}$-modules on a smooth, proper curve, over an algebraically closed field $k$ of characteristic $p > 0$, as a sum of local and global terms, where $l \ne p$. The primary focus is on removing the restriction on $l$. We begin with calculations for $p$-torsion sheaves trivialized by $p$-extensions, but using etale cohomology to give a unified proof for all primes $l$. In the remainder of this work, only $p$-torsion sheaves are considered. We show the existence on ${X_{{\text {et}}}}$, $X$ a scheme of characteristic $p$, of a short exact sequence of sheaves, involving the tangent space at the identity of a finite, flat, height 1, commutative group scheme, and the subsheaf fixed by the $p$th power endomorphism; the latter turns out to be an etale group scheme. A corollary gives complete results on the Euler-Poincaré characteristic of a constructible sheaf of ${F_p}$-modules on a smooth, proper curve, over an algebraically closed field $k$ of characteristic $p > 0$, when the generic stalk has rank $p$. Explicit computations are given for the Euler characteristics of such $p$-torsion sheaves on ${P^1}$ and a result on elliptic surfaces is included. A study is made of the comparison of the $p$-ranks of abelian extensions of curves. Several examples of $p$-ranks for nonhyperelliptic curves are discussed. The paper concludes with a brief sketch of results on certain constructible sheaves of ${F_q}$-modules, $q={p^r}, r \ge 1$.