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 {F_l} -modules on a smooth, proper curve, over an algebraically closed field k k of characteristic p > 0 p > 0 , as a sum of local and global terms, where l ≠ p l \ne p . The primary focus is on removing the restriction on l l . We begin with calculations for p p -torsion sheaves trivialized by p p -extensions, but using etale cohomology to give a unified proof for all primes l l . In the remainder of this work, only p p -torsion sheaves are considered. We show the existence on X et {X_{{\text {et}}}} , X X a scheme of characteristic p 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 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 {F_p} -modules on a smooth, proper curve, over an algebraically closed field k k of characteristic p > 0 p > 0 , when the generic stalk has rank p p . Explicit computations are given for the Euler characteristics of such p p -torsion sheaves on P 1 {P^1} and a result on elliptic surfaces is included. A study is made of the comparison of the p p -ranks of abelian extensions of curves. Several examples of p p -ranks for nonhyperelliptic curves are discussed. The paper concludes with a brief sketch of results on certain constructible sheaves of F q {F_q} -modules, q = p r , r ≥ 1 q={p^r},\,r \ge 1 .