On the semisimplicity of pure sheaves

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.