# If one wants to work with $Q_l$-adic sheaves, should the scheme be of finite type over a 1-dimensional one?

In section 6 of his 'Adic Formalism' T. Ekedahl states that $l$-adic sheaves 'behave nicely' for finite type separated schemes over $S$ that is regular of dimension $\le 1$. Is the dimension restriction really necessary here? A similar restriction is used by A. Huber when she studies 'arithmetic perverse sheaves'.

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## 2 Answers

Gabber has announced a proof of the finiteness theorem for (direct images of constructible sheaves by) morphisms of finite type between general noetherian schemes years ago, but, being Gabber, he has not written it down. There was a seminar in the Ecole Polytechnique four years ago (I think) about his proof, and the goal was to write a full proof (probably a book), but it doesn't look as if they are done yet. You can find statements and outlines of proofs in notes on Illusie's webpage (e.g., http://www.math.u-psud.fr/~illusie/Illusie_Tokyo.pdf), there are some notes about at least parts of the proofs and useful statements on Joël Riou's webpage (http://www.math.u-psud.fr/~riou/), I think some of the ideas are showing up in fabrice Orgogozo's work (for example, Orgogozo announced a result about uniform constructibility for torsion sheaves and I'm pretty sure it uses Gabber's ideas, but I don't find a preprint on his webpage). Actually, I think Gabber's uniformization theorem has been used in other works; maybe Frederic Deglise's work ?

(Oh, and all the references are in French. Enjoy. ;-))

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Thank you very much! Since this matter is not really crucial for me, I will restrict myself to the 1-dimensional base, and make a remark containig your information. By the way, I would prefer to put your (full) name in this remark.:) –  Mikhail Bondarko Feb 4 '11 at 22:36

At the time of the writing of the 'Adic formalism' the complete finiteness and the formalism of all the relevant operators was only known for the case mentioned (I think that SGA 4 1/2 was the most complete reference). I don't think there is any reason to believe that the results wouldn't be true in more generality but I am not up enough on the current literature to know whether more general results are now known.

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Thank you very much for such a comprehensive explanation of your assumptions! –  Mikhail Bondarko Feb 5 '11 at 10:23