# cohomological dimension of the push-forward functor

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 $\ell$).

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I suppose $f$ is separated. Read Deligne's marvelous "Th. finitude" in SGA 4.5 (source of all ideas in the K-W argument). Deligne's inductive methods prove that $f_{\ast}$ has cohom. dimension (on torsion-sheaves with torsion-orders invertible on the regular base of dim. at most 1) at most $2 {\rm{dim}}(X) + {\rm{dim}}(Y)$. For the analytic version $2 {\rm{dim}}(X)$ works assuming $X$ paracompact & Hausdorff: use topological dim. theory & link of higher direct images with sheafified cohom. & link of derived functor cohom. with Cech theory on paracompact Hausdorff spaces. –  BCnrd Oct 1 '10 at 17:16
Thanks, Brian. Now I seem to remember you told me this before...haha. Do you think the proof of KW is correct? It only refers to SGA4, Exp X. Maybe I missed something, but it doesn't make any sense to me. BTW, I think we can remove the separated assumption. For a general morphism f, let U be an open affine in X with complement Z. Then U is separated over Y. Consider the exact triangles f_*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
Dear Shenghao: well, when I presented this result in the Weil II seminar a couple of years ago I didn't see any problems with K-W, perhaps in part because anything which was unclear I knew how to handle by looking back at Deligne's article. What you say about removing separatedness sounds superficially OK, but I haven't sat down to think it through carefully. –  BCnrd Oct 1 '10 at 22:31