Question

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?

Answer 1

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.

Answer 2

Actually, the proof of Remark 8.6 is elementary. Let $\alpha$ be an endomorphism of vector space $W$, and suppose that it doesn't act semisimply. Then there exists a vector $e$ and a scalar $a$ such that $(\alpha-a)^2e=0$ but $(\alpha-a)e\neq 0$, say $\alpha e=ae+t$ with $t\neq 0$ (after possibly extending scalars). In the situation of 8.6, we have another vector space $W$, endomorphism $\beta$, and a vector $f$ such that $\beta f=a^{-1}f$. Now one checks that $(\alpha\otimes\beta-1)^2(e\otimes f)=0$ but $(\alpha\otimes\beta-1)(e\otimes f)\neq 0$, contradicting the semisimplicity of $\alpha\otimes\beta$ at $1$.