Continuity of l-adic cohomology: is the cohomology of the generic point isomorphic to the completion of the limit of cohomology of open subvarieties?

Question

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.

Answer 1

You may consider $X=\mathbf{P}^1_k$, and look at the first cohomology group $\varinjlim H^1_{et}(X_i,\mathbf{Z}_l)$. By the toposic version of Hurewicz theorem, this colimit is the same as $\varinjlim Hom_{cont}(\pi_1(X_i),\mathbf{Z}_l)$ (where $Hom_{cont}$ means continous morphisms of topological groups). Each $X_i$ is $X$ minus a finite set of closed points, so that $\varinjlim H^1_{et}(X_i,\mathbf{Z}_l)$ will look very much like a free $\mathbf{Z}_l$-module on an infinite set (once $\infty$ has been removed, we freely add a generator each time we leave out some point; take $k$ to be the field of complex numbers to fix the ideas). On the other hand, we can do the same computation with finite coefficients, and thus, using the continuity of étale cohomology with finite coefficients, conclude that the $l$-adic Galois cohomology of $\eta$ is (or, at the very least, surjects to) the $l$-adic completion of $\varinjlim H^1_{et}(X_i,\mathbf{Z}_l)$. This clearly provides a counterexample.

As for the question if an open immersion can induce an injective map on cohomology, you can again look at an open subset of a smooth curve: using the Gysin long exact sequence, you will see that the cohomology of the complement has to fit somewhere and that this gives serious obstructions to injectivity (at least if you want such a property in every degree).