I expect that if $X$ is a finite type $k$-scheme, $k$ a field of characterisic $p$, then for primes $\ell\neq p$, $H^*_{cdh}(X;\mathbb{Z}/\ell\mathbb{Z})\cong H^*_{h}(X;\mathbb{Z}/\ell\mathbb{Z})$. Is this true? If so, can someone describe the philosophy behind why such cohomological comparison theorems with suitable finite coefficients are expected to be true? Also, references for this particular theorem and possible generalizations would be very nice.
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12$\begingroup$ This can fail already for $X=\operatorname{Spec}(k)$: fields have no cdh cohomology, but $H^*_h(X,\mathbb Z/l)$ always agrees with etale cohomology. A reference for the latter fact is Voevodsky's thesis math.uiuc.edu/K-theory/0031/H1.pdf. $\endgroup$– Marc HoyoisCommented Sep 16, 2016 at 2:04
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