# Do all the main properties of constructible and perverse sheaves (in an 'arithmetic' situation) follow from results of Gabber?

This question is a continuation of 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 My question is: are the recent unpublished results of Gabber (see here) sufficient for all of this?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!!

-
I've fixed the LaTeX and links; I'm not sure why, but recently (I don't remember this always having been the case) I've been finding that one needs to put a backslash before underscores for them to be parsed correctly. Also I'm pretty sure the actual * was causing problems because it was being interpreted as the markdown command for italicized text - I replaced it with the LaTeX command \ast. –  Zev Chonoles Mar 11 '11 at 13:19
Thank you very much!!! –  Mikhail Bondarko Mar 11 '11 at 13:27
Mikhail, I'll be waiting for you to write this up! –  shenghao Mar 11 '11 at 16:16
Actually, I was not going to write that up; I just wanted to say that this is easy.:) It seems that either this works (and so everything was actually done by Gabber), or I miss something important in this picture (and then I would hardly be able to close the gap). –  Mikhail Bondarko Mar 11 '11 at 16:34
Yes, that is all true if you work with quasi-excellent schemes. In fact, this is even true for $\mathbf{Z}_\ell$ coefficients. For $\mathbf{Q}_\ell$ coefficients, if you work with schemes whose $\ell$-cohomological dimension at each residue field is uniformly bounded (e.g. of finite type over \mathbf{Z}$) you can prove that$Rf_*\$ preserves constructible sheaves of geometric origin using only de Jong's resolution of singularities by alterations, as absolute purity with rational coefficients was then proved by Thomason; see Theorem 4.1 in numdam.org/item?id=ASENS_1985_4_18_3_437_0 –  Denis-Charles Cisinski Mar 11 '11 at 19:26