This question is a continuation of http://mathoverflow.net/questions/55790/bad-behaviour-of-perverse-sheaves-over-general-bases

Let $S$ (for example) be a finite type separated scheme over $\mathbb{Z}$. I would like: (1) to define the perverse $t$-structure for the derived category of etale ${\mathbb{Q}}\_l$  sheaves over $S$; (2) ${f}\_{\ast}$ to be $t$-exact if $f$ is quasi-finite affine, and (3) $f^{\ast}[d]$ to be $t$-exact if $f$ is smooth of relative dimension $d$. My question is: are the recent unpublished results of Gabber (see [here][1]) sufficient for all of this?

As far as I understand, in order to define the perverse $t$-structure using stratifications (possibly, this is not the best approach) one needs: (1a) for finite type $f$ the functors $f^{{\ast},!}$ and $f_{{\ast},!}$ should respect constructibility + (1b) relative purity. (1a) in my case seems to be well-known, and was proved by Gabber in a very general situation. Whereas in BBD (in the case when $S$ is variety) SGA4.XVI.3.7 was used for (1b), it seems that Gabber's purity result can replace loc.cit. in our situation.  Next, it seems that (2) follows from Affine Lefschetz (see section 4 of [1]) along with Verdier duality. Lastly, (3) seems to be straightforward from the definition of the middle perversity.


Is all of this true?:) Did I miss anything important? I would be deeply grateful for any comments!! 





  [1]: http://www.math.u-psud.fr/~illusie/Illusie_Fields.pdf