I'm trying to find a reference for the following fact:



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> If Tate's conjecture is true for all smooth projective varieties over $\mathbb{F}\_p$, then the Frobenius endomorphism on the crystalline cohomology of any such variety is semisimple.
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This is stated in the Coleman-Edixhoven paper on the semisimplicity of the $U\_p$-operator on modular forms. They reference Milne's paper "Motives over finite fields" (in the 1991 Motives conference proceedings, ed. Janssen/Kleiman/Serre). 


I found Milne's paper on the web, and it gives two references for the corresponding statement for $\ell$-adic cohomology ($\ell \ne p$) and then says "There is an analogous statement ... for the crystalline cohomology" without giving a reference (or a precise statement) for this. Moreover, one of the references for the $\ell \ne p$ case is to Milne's book "Arithmetic Duality Theorems" but points to an apparently non-existent section 8.6; while the other reference Milne gives is to Tate's article in the same proceedings, which does not seem to prove anything about semisimplicity as far as I can see. 


Can anyone tell me where I can find a proof of the above implication written down? 



Why does Tate's conjecture imply semisimplicity of crystalline Frobenius?
Milne's Remark 8.6 in Amer. J. Math. 1986 implicitely includes two algebraic statements:


1. Tensor product respects generalised eigenspaces.
2. Let $V, W$ be representations of a group over a field of characteristic $0$. If $V\otimes W$ is semi-simple, then $V$ is semi-simple.


Both statements are true but not so immediate. A proof of 2 is in Serre, J. Alg. 194 (1997), Prop. 2.3, while his prop. 7.2.1 shows that characteristic $0$ is necessary (in char. $p$, $\dim W\not\equiv 0\pmod{p}$ is sufficient but the other congruence can yield counterexamples).


For 1, here is a short argument. Let $v$ an endomorphism of $V$ and $w$ be an endomorphism of $W$. Assume that $(v-a)^m=0$ and that $(w-b)^n=0$ for some scalars $a,b$. Then 
$$(v\otimes w - ab)^{mn}=0$$
because $v\otimes w - ab=(v-a)\otimes w + a1\otimes(w-b)$, a sum of two nilpotent operators which commute.


