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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On the semisimplicity of pure sheaves

Author: Lei Fu
Journal: Proc. Amer. Math. Soc. 127 (1999), 2529-2533
MSC (1991): Primary 14F20, 14G15
Published electronically: May 19, 1999
MathSciNet review: 1676348
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Abstract: We obtain a criteria for a pure sheaf to be semisimple. As a corollary, we prove the following: Let $X_0$ and $S_0$ be two schemes over a finite field $\mathbf{F}_q$, and let $f_0: X_0\rightarrow S_0$ be a proper smooth morphism. Assume $S_0$ is normal and geometrically connected, and assume there exists a closed point $s$ in $S_0$ such that the Frobenius automorphism $F_s$ acts semisimply on $H^i(X_{\bar s}, {\overline{\mathbf{Q}_l}})$, where $X_{\bar s}$ is the geometric fiber of $f_0$ at $s$ (this last assumption is unnecessary if the semisimplicity conjecture is true). Then $R^if_{0\ast} {\overline{\mathbf{Q}_l}}$ is a semisimple sheaf on $S_0$. This verifies a conjecture of Grothendieck and Serre provided the semisimplicity conjecture holds. As an application, we prove that the galois representations of function fields associated to the $l$-adic cohomologies of $K3$ surfaces are semisimple. We also get a theorem of Zarhin about the semisimplicity of the Galois representations of function fields arising from abelian varieties. The proof relies heavily on Deligne's work on Weil conjectures.

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Lei Fu
Affiliation: Institute of Mathematics, Nankai University, Tianjin, People’s Republic of China

Received by editor(s): August 11, 1997
Published electronically: May 19, 1999
Communicated by: Ron Donagi
Article copyright: © Copyright 1999 American Mathematical Society