Let $X$ be a variety over an algebraically closed field $k$. Denote by $\eta$ its generic point; it is the inverse limit of the open subvarieties $X_i$ of $X$. It is well known that the etale cohomology of $\eta$ (i.e. the corresponding cohomology of the Galois group of $\eta$) with $\mathbb{Z}/l^n\mathbb{Z}$-coefficients is isomorphic to the direct limit of the $\mathbb{Z}/l^n\mathbb{Z}$-cohomology of $X_i$ (here $l\ne char\, k$, $n>0$). I would like to know: is it true that $H^\ast_{et}(\eta, \mathbb{Z}_l)$ is the $l$-adic completion of $\varinjlim H^\ast_{et}(X_i, \mathbb{Z}_l)$ (here one should consider the continuous etale cohomology of $\eta$ i.e. the continuous $\mathbb{Z}_l$-cohomology of its Galois group, whereas for $X_i$ the continuous etale cohomology is just the 'naive' one)? This seems to be equivalent to: $\varinjlim H^\ast_{et}(X_i, \mathbb{Z}_l)$ contains no infinitely divisible elements. What can one say about this limit? A related question: when does an open dense embedding of varieties necesarily yield an embedding of their $\mathbb{Z}/l\mathbb{Z}$-cohomology?

The problem is that the cohomology of $\eta$ is usually not finitely generated. So I was not able even to find the answer for the corresponding continiuos profinite group cohomology (of the fundamental groups of $X_i$) question (whereas I am not sure at all that one can replace $\varinjlim H^\ast_{et}(X_i, \mathbb{Z}_l)$ by $\varinjlim H^\ast_{cont}(\pi_1(X_i), \mathbb{Z}_l)$ here).

PS. I was silly not to complete $\varinjlim H^\ast_{et}(X_i, \mathbb{Z}_l)$ in the first version of my question.