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

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.

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?

They reference Milne's paper...Hasreferencebecome a verb in England too? – Chandan Singh Dalawat Feb 21 '11 at 5:56citeexamples, or do theyreferencethem? (Sorry, I could not stop myself.) – jmc Nov 6 '12 at 0:26