Is it easy to define weights for $Q_l$-sheaves over finite type $Z[1/l]$-schemes?

In her paper "Mixed perverse sheaves for schemes over number fields" (http://www.springerlink.com/content/w848j73552102274/) A. Huber defines certain weights for certain categories of $\mathbb{Q}_l$-sheaves over a finite type $\mathbb{Q}$-scheme $S$ using the following argument:

she presents $S$ as a limit of finite type $U_0$-schemes $S_0$, where $U_0$ is an open subscheme of $\mathbb{Z}[1/l]$. Then (as far as I understand) she defines her (derived) category of sheaves over $S$ as a 2-limit of those for all $S_0$. She states that there exist a perverse $t$-structure and a nice notion of weights for this limit category that could be obtained as the limit of the corresponding structures over all $S_0$. So, it seems that the corresponding properties of sheaves and weights over $S_0$ (see Huber's 3.4-3.9) should be true (and the proof is even easier; Huber just relies on the corresponding results of BBD). Is this correct, or is there some matter that simplifies when we pass from $S_0$ to $S$?

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The answer to the question in your title is, I think : "in general, no". The answer to your last question is : "well, it depends how you have defined the objects, and you will have to be very careful about which morphisms of schemes you allow".

Let me be more precise. Huber wants to consider only complexes of sheaves on $S_0$ that are constructible with respect to a stratification all of whose strata are smooth over $U_0$ (I will explain why later). But direct images by morphisms of $U_0$-schemes do not respect this kind of constructibility in general, they only respect it after you restrict to an open subscheme of $U_0$, so she has to do that and take the direct limit over all open subschemes of $U_0$. See the end of section 2, where she explains that you could stay on $U_0$ if you put supplementary conditions on the morphisms of schemes (for example, proper and smooth morphisms between smooth $U_0$-schemes will be fine).

Now here are the reasons why she chooses this restrictive notion of constructibility :

(1) She needs it to show that her definition of the perverse $t$-structure works. See the remark after theorem 2.5. I am not sure that this is necessary, because now we have more general ways to construct $t$-structures on categories of complexes of sheaves (see Gabber's "Note on some $t$-structures"), although there is still the problem of checking that your category of perverse sheaves satisfies the usual conditions (finiteness, description of simple objects etc) or of showing that you don't really need these things.

(2) She defines weights by restricting her complexes over the fibers over closed points of $U_0$, and she wants to prove all the properties of weights by citing BBD (Beilinson-Bernstein-Deligne, Astérisque 100). To do that, she needs the fact that restriction to fibers over closed points of $U_0$ will commute with the 4 (or 6, or 7) operations on her categories of complexes, and this would not be true for more general complexes. What she does, actually, is just to cite Deligne's generic base change theorem, which will immediately give you the compatibility you want, but at the cost of shrinking the base (i.e., of shrinking $U_0$).

(3) Related to (2) : Given the way she defines weights, she wants her weights to be the same over each closed point of $U_0$. That is of course not going to be true if she allows stratifications with strata that are not necessarily smooth over $U_0$ : you could have a sheaf that is concentrated over one closed point of $U_0$, for example.

I am not saying that it is necessary to do what she does, just that it makes your life much simpler. It might be possible to develop a theory of weights for general constructible sheaves (i.e., constructible with respect to a general stratification). But you will not be able to reduce everything to BBD anymore because you won't be able to use generic base change, and you will likely run into very hard problems like the weight-monodromy conjecture when you try to prove that the 4 operations preserve mixed sheaves and that they have the expected effect on weights.

Now if you're looking at motives instead of $\ell$-adic complexes, the situation is very different because your approach is different : all motives are already mixed, the fact that, say, $f_*$ increases weights is somehow built in the very definition of weights, etc. Then the problem is of course to related weights for motives and weights for $\ell$-adic complexes. Note that the weight-monodromy conjectures follows from the "standard" conjectures of motives, so if we had the full formalism of motives all our problems would probably disappear (it doesn't much help us at this point but it might at least make us feel better to know that the world is conjectured to be coherent).

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Thank you very much! This means that if one starts from constant sheaves, and applies only those morphisms that are described in Huber's Proposition 2.11, then the weights are still OK? Hope this will be sufficient for me. Also, did anybody try to write down a 'full' version of weights for arithmetic sheaves (that relies on some conjectures)? –  Mikhail Bondarko Feb 8 '11 at 17:41
First question : yes, I think so. Second question : not as far as I know. I think that, even if everything else worked, you would still run into the problem that mixed perverse sheaves don't necessarily have a weight filtration. (This problem appears in Huber's work too.) –  Alex Feb 8 '11 at 17:54
Luckily, I don't need a weight filtration for all mixed sheaves. If I would have functoriality, then for sheaves of 'motivic origin' it would exist automatically (it is given by the corresponding weight spectral sequences). –  Mikhail Bondarko Feb 8 '11 at 19:18
Besides, I really owe you a mentioning in my papers!:) –  Mikhail Bondarko Feb 8 '11 at 19:55