# Etale cohomology of a variety over a number field VS of its $p$-adic completion

Let $$X$$ be a quasi-projective variety over a number field $$E$$. Let $$v$$ be a place in $$E$$ and let $$E_v$$ denote the $$v$$-adic completion. Let $$p$$ be the prime number determined by $$v$$, and let $$\Lambda = \mathbb Z_{\ell}/\ell^k\mathbb Z_{\ell}$$, where $$\ell$$ is a prime number different from $$p$$ and $$k\geq 1$$.

Under what conditions do we have an isomorphism between the etale cohomology of $$X/E$$ and the etale cohomology of $$X_v/E_v$$, both with coefficients in $$\Lambda$$?

In particular, I am interested in the case where $$v$$ is non-archimedean.

You shouldn't expect such an isomorphism in general, even if $$X$$ is projective. When $$X=\mathrm{Spec}(E)$$, you are asking whether $$H^i(E,\mathbb Z/\ell^k)\cong H^i(E_v,\mathbb Z/\ell^k)$$.
For example when $$E$$ contains a $$\ell$$-th root of unity, then $$\mathbb Z/\ell\cong\mu_\ell$$ as Galois modules, and by Hilbert's 90, the short exact sequence $$1\to\mu_\ell\to\mathbb G_m\xrightarrow{\ell}\mathbb G_m\to 1,$$ so that $$1\to \mu_\ell\to E^\times\xrightarrow\ell E^\times\to H^1(E,\mu_\ell)\to 1,$$ so that $$H^1(E,\mu_\ell)\cong E^\times/(E^\times)^\ell$$ while $$H^1(E_v,\mu_\ell)\cong E_v^\times/(E_v^\times)^\ell$$. These are far from isomorphic in general: $$E^\times/(E^\times)^\ell$$ is infinitely-generated, while $$E_v^\times/(E_v^\times)^\ell$$ is certainly finite.