Let $\ell$ be a prime number and let $f:X \to Y$ be a morphism of schemes of finite type over the complex numbers (or a regular scheme of dimension at most 1, in which $\ell$ is invertible). How to prove that the $\ell$-cohomological dimension of $Rf_\*$ in the etale topology is finite? I don't understand the proof in Kiehl and Weissauer's book "Weil Conjectures, Perverse Sheaves and l’adic Fourier Transform", Appendix D, since there is no base change for $f_*$. The crucial case seems to be when $f$ is an open immersion.

And what about the analytic version? $f:X \to Y$ is a morphism of complex analytic spaces, and $F$ is a sheaf of abelian groups on $X$ (in particular, no $\ell$).

j_!j^*F \to f_*F \to (fi)_*i^*F and f_*j_!G \to (fj)_*G \to (fi)_*i^*j_*G, and use the coh. dim. of j_, where the open immersion j is separated.... – shenghao Oct 1 '10 at 22:07