# Help with $\mathbf{Q}_{\ell}$ sheaves

Let $$X\to S$$ be a morphism of smooth connected varieties over an algebraically closed field $$k$$; let $$j:\eta\to S$$ be the inclusion of the generic point into $$S$$ (not a geometric generic point) and let $$\mathscr F$$ be a constructible étale sheaf of $$\Lambda$$-modules on $$X_\eta$$. If $$\Lambda=\mathbf{Z}/\ell^n\mathbf{Z}$$, for example, then (letting $$j$$ also denote its base extension $$X_\eta\to X$$), $$j_*\mathscr F$$ is a constructible sheaf. But if $$\Lambda=\mathbf Z_\ell$$ or $$\mathbf Q_\ell$$, is $$j_*\mathscr F$$ still constructible? When I think about applying $$j_*$$ to a projective system (of constructible sheaves of modules for finite coefficients), the result is a projective system of constructible sheaves. When I think about it when $$X=S$$ is a smooth curve and think about $$\mathscr F$$ as a $$\mathbf Q_\ell$$-representation $$V$$ of $$\operatorname{Gal}(\overline\eta/\eta)$$, however, then the condition is equivalent to the condition that the inertia of all but finitely many points of $$S$$ act trivially on $$V$$—this seems dodgy. Thanks for your help.

A necessary condition is that for each stratum $$X_{\eta}^i$$ of some stratification of $$X_{\eta}$$ on which $$\mathcal F$$ is lisse, the associated representation of $$\pi_1(X_{\eta}^i)$$ is unramified away from a closed subset $$Z^i$$ of the closure $$\overline{X_\eta^i}$$ of $$X_{\eta}^i$$ in $$X$$, such that the projection of $$Z_i$$ to $$S$$ is not dense.
This can indeed fail, so the very general statement you wrote is not true. Your idea is precisely the right one - it's easy to construct a Galois representation that is not finitely ramified, for instance by constructing a suitable extension of $$\mathbb Z_\ell$$ by $$\mathbb Z_\ell(1)$$ using Kummer theory.
What will go wrong in the limiting proof you want to write down is that the pushforward of the associated $$\mathbb Z/\ell^n$$ pushforward, modulo $$\ell$$, will not be the pushforward of the associated $$\mathbb Z_\ell$$-sheaf, and will not even stabilize - it will be a sheaf with zero stalk at more and more ramification points as $$n$$ goes to $$\infty$$.
I think this condition is also suffiicent, but I didn't check it outside the case where $$X =S$$, where it can be proven by first pushing forward to the open set where the Galois representation is unramified, and then pushing tot he whole space.