Let $X$ be a smooth projective variety over a field $k$. Suppose we have étale abelian sheaves $A, B$ on $X_{\rm ét}$ such that
$$H^j(X_{\rm ét}, A),\ H^j(X_{\rm ét}, B)$$ are finitely generated abelian groups for all $j$. Is $\mathbb{H}^j(X_{\rm ét}, A\otimes^L_{\mathbf{Z}} B)$ finitely generated?
No. Take a sheaf over $\mathbb F_\ell$ whose etale cohomology vanishes (for instance, $X$ an elliptic curve over an algebraically closed field, $\mathcal L$ the locally constant sheaf on $X$ associated to a nontrivial one-dimensional representation of its fundamental group into $\mathbb F_\ell^\times$ ), let $A$ be the sum of infinitely many copies of the sheaf, and let $B$ be the sum of infinitely many copies of the dual sheaf. Then the etale cohomology of $A$ and $B$ both vanish (because $\mathbb F_\ell^\infty$ is flat and tensoring with it preserves cohomology), but their tensor product is the sum of infinitely many copies of the constant sheaf, which has nonvanishing cohomology.
