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Transactions of the American Mathematical Society

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

Author: William Anthony Hawkins
Journal: Trans. Amer. Math. Soc. 301 (1987), 163-188
MSC: Primary 14F20; Secondary 14L15
MathSciNet review: 879568
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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$.

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