On the semisimplicity of pure sheaves

We obtain a criteria for a pure sheaf to be semisimple. As a corollary, we prove the following: Let and be two schemes over a finite field , and let be a proper smooth morphism. Assume is normal and geometrically connected, and assume there exists a closed point in such that the Frobenius automorphism acts semisimply on , where is the geometric fiber of at (this last assumption is unnecessary if the semisimplicity conjecture is true). Then is a semisimple sheaf on . 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 -adic cohomologies of 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.