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Let $k$ be a field, $X$ a smooth $K$-variety and $\ell$ a prime not dividing the characteristic of $K$.

Then one can make sense of continuous $\ell$-adic etale cohomology (in the sense of Jannsen), both of $X$ and of the base-change $X_{\overline{K}}$ to the separable closure, and there's a spectral sequence $$ E^{pq}_2 = H^p(K, H^q_{\mathrm{et}}(X_{\overline{K}}, \mathbf{Q}_\ell)) \Rightarrow H^{p + q}_{\mathrm{et}}(X, \mathbf{Q}_\ell). $$

According to this paper by Nekovar, the spectral sequence is known to degenerate at $E_2$ if $X$ is proper, as a consequence of various deep results of Deligne.

Are there examples of smooth (but non-proper) $X$ where this sequence really doesn't degenerate at $E_2$? I'm particularly interested in the case where the base field $K$ is a number field or a $p$-adic field.

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  • $\begingroup$ Just out of curiosity: the $d_2$-differential goes $H^p(K,H^q_{\textrm{et}}(X_{\overline{K}}))\to H^{p+2}(K,H^{q-1}_{\textrm{et}}(X_{\overline{K}}))$. So for finite fields the spectral sequence would always degenerate at $E_2$ because of cohomological dimension 1. For $p$-adic fields, it would automatically degenerate at $E_3$ because of cd $2$. The finite fields statement could guide the search for examples over $p$-adic fields, they have to make use of cohomological dimension $2$. $\endgroup$ May 19, 2014 at 16:17
  • $\begingroup$ I agree: over $p$-adic fields it's all about the vanishing of the differentials $H^0(K, H^q) \to H^2(K, H^{q-1})$, but I don't how to use this to manufacture a counterexample. Part of the motivation for the question is that I don't have the foggiest idea what the absolute etale cohomology will look like if the sequence doesn't degenerate at E_2! $\endgroup$ May 19, 2014 at 19:22

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