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 $l\neq p$?
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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’EulerPoincaré en cohomologie $\ell$adique, C. R. Acad. Sc. Paris, t. 292 (1981), Série I, 209212 that for $X$ a separated scheme of finite type over $k$ a field of characteristic $p>0$, the EulerPoincaré characteristic (over $\bar{k}$) is independent of $\ell≠p$ (the proof consists in showing that the EulerPoincaré characteristic coincides with the EulerPoincaré characteristic in compact support; the latter being the degree of the Zeta function and thus independent of $\ell$). Hope this helps. 

