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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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The Ă©tale cohomology of $p$-torsion sheaves. I
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by William Anthony Hawkins PDF
Trans. Amer. Math. Soc. 301 (1987), 163-188 Request permission


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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Additional Information
  • © Copyright 1987 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 301 (1987), 163-188
  • MSC: Primary 14F20; Secondary 14L15
  • DOI:
  • MathSciNet review: 879568