The étale cohomology of -torsion sheaves. I
Author:
William Anthony Hawkins
Journal:
Trans. Amer. Math. Soc. 301 (1987), 163-188
MSC:
Primary 14F20; Secondary 14L15
DOI:
https://doi.org/10.1090/S0002-9947-1987-0879568-5
MathSciNet review:
879568
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Abstract | References | Similar Articles | Additional Information
Abstract: This paper generalizes a formula of Grothendieck, Ogg, and Shafarevich that expresses the Euler-Poincaré characteristic of a constructible sheaf of -modules on a smooth, proper curve, over an algebraically closed field
of characteristic
, as a sum of local and global terms, where
. The primary focus is on removing the restriction on
. We begin with calculations for
-torsion sheaves trivialized by
-extensions, but using etale cohomology to give a unified proof for all primes
.
In the remainder of this work, only -torsion sheaves are considered. We show the existence on
,
a scheme of characteristic
, 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
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
-modules on a smooth, proper curve, over an algebraically closed field
of characteristic
, when the generic stalk has rank
.
Explicit computations are given for the Euler characteristics of such -torsion sheaves on
and a result on elliptic surfaces is included. A study is made of the comparison of the
-ranks of abelian extensions of curves. Several examples of
-ranks for nonhyperelliptic curves are discussed. The paper concludes with a brief sketch of results on certain constructible sheaves of
-modules,
.
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DOI:
https://doi.org/10.1090/S0002-9947-1987-0879568-5
Article copyright:
© Copyright 1987
American Mathematical Society