Let $X$ be a proper variety over a finite field $k$ of characteristic $p>0$, and let $\mathcal F$ be a finite rank $\mathbb F\_\ell$ local system on (the etale site of) $X$. Is it true (and, if so, how does one prove) that
$$\chi(X,\mathcal F)=\operatorname{rk}\mathcal F\cdot\chi(X,\underline{\mathbb F\_\ell}\_X)$$
where $\underline{\mathbb F\_\ell}\_X$ denotes the constant local system on $X$?


Some remarks:


* If $k=\mathbb C$, then this is true (without assuming properness), but the only proof I know is via analytification and topological arguments. This implies the result for $(X,\mathcal F)$ which lift to characteristic zero, but this leaves out lots of varieties $X$ and local systems $\mathcal F$.
* Properness is necessary when $\operatorname{char}k>0$. Namely, let $f:\mathbb A^1\to\mathbb A^1$ denote the Artin--Schrier map over $\mathbb F\_p$, so $f\_\ast\underline{\mathbb F\_\ell}\_{\mathbb A^1}$ is a local system since $f$ is etale. Now $\operatorname{rk}f\_\ast\underline{\mathbb F\_\ell}\_{\mathbb A^1}=p$, but $\chi(\mathbb A^1,f\_\ast\underline{\mathbb F\_\ell}\_{\mathbb A^1})=\chi(\mathbb A^1,\underline{\mathbb F\_\ell}\_{\mathbb A^1})$ by the Leray spectral sequence.



Euler characteristic of local system depends only on rank?
As $X$ is proper, the Swan conductor of $\mathcal{F}$ vanishes. Hence the identity $\chi(X,\mathcal F)=\operatorname{rk}\mathcal F\cdot\chi(X,\underline{\mathbb F\_\ell}\_X)$ follows from Theorem $4.2.9$ in Kato & Saito's ["Ramification theory for varieties over a perfect field
".](https://arxiv.org/abs/math/0402010)


