In Weil II, Deligne proves that the six functors preserve the category of mixed $l$-adic complexes. Most difficulties are encountered when proving that if $f\colon X\to Y$ is a finite type separated morphism over the algebraic closure of a finite field, then $Rf_!$ and $Rf_*$ send mixed complexes to mixed complexes.
My question is: is there a direct way to see this in the case of an open embedding $j\colon U\to X$? Given a mixed complex (or just a mixed sheaf) $K$ on U, what do we need to know to be able to show that the complexes $Rj_!K$ and $Rj_*K$ are mixed?
For example if $F$ is a punctually pure sheaf, it seems clear that $j_!F$ is punctually pure, simply beacuse its stalks are $0$ outside $U$ and coincide (as Galois modules) with those of $F$ at the fixed points inside $U$. Now say that $X$ is a curve.
Then the complement $i\colon Z\to U$ is a finite set of points; a complex on $Z$ is mixed if and only if all Frobenius eigenvalues have norm a power of $q$, where $q$ is the cardinality of the base field. Hence, thanks to the exact sequence of sheaves $$0\to j_!F\to j_*F\to j_*F/j_!F\to 0, $$ to show that $j_*F$ is mixed it suffices to show that the eigenvalues of Frobenius on $j_*F/j_!F$ are of absolute values a power of $q$. Am I missing something here?
