I am interested in (Chow) motives over (algebraically closed) characteristic $p>0$ fields. For $H$ being $\mathbb{Q}\_l$-adic cohomology, one can consider $Ch\_l(M)=\sum (-1)^i\dim\_{\mathbb{Q}\_l} H^i(M)$.
Does this Euler characteristic depend on the choice of $l\neq p$?



Do $Q_l$-etale Euler characteristics of Chow motives coincide for all $l$?
I might be missing something (seeing that I don't know what a Chow motive is), but I think the answer is yes. It is a result of G.Laumon proved in Comparaison de caractéristiques d’Euler-Poincaré en cohomologie $\ell$-adique, C. R. Acad. Sc. Paris, t. 292 (1981), Série I, 209-212 that for $X$ a separated scheme of finite type over $k$ a field of characteristic $p>0$, the Euler-Poincaré characteristic (over $\bar{k}$) is independent of $\ell≠p$ (the proof consists in showing that the Euler-Poincaré characteristic coincides with the Euler-Poincaré characteristic in compact support; the latter being the degree of the Zeta function and thus independent of $\ell$). Hope this helps.


