I would like to understand the theory of pure complexes of (etale?) sheaves (of geometric origin?). In particular, I would like to understand which conditions are realy necessary in (part 1 of) Theorem 3.1.8 of Cataldo-Migliorini's survey <http://www.ams.org/journals/bull/2009-46-04/S0273-0979-09-01260-9/S0273-0979-09-01260-9.pdf>
(see page 33-567). Does the splitting of part 1 (only!) really requires $\overline{\mathbb{Q}\_l}$-coefficients? Which coefficients could be put here? Does the splitting exist over $X\_0$?


Below they explain that those extensions of mixed complexes $K\_0,L\_0$ of appropriate weights that come from $\mathbb{F}\_q$, become zero over $\mathbb{F}$. My main question is: does there exist a triangulated category of complexes of sheaves where the corresponding Ext-group is zero from the beginning (i.e. we consider $Ext^1$ in a single triangulated category instead of the image $Ext^1(K\_0.L\_0)\to Ext^1(K,L)$). Is there such a category over a (more or less) general base scheme $S$ (instead of $\mathbb{F}\_q$ or $\mathbb{F}$)? Again, which coefficient rings are possible here?



Morphisms between pure complexes of sheaves
Dear Mikhail,


I had been hoping someone else would attempt to answer this question, as I have been wondering very similar things lately. (In fact I drove myself crazy for about a month last year trying to work out some solution to what you are asking in the second paragraph.)


I can't answer everything but here is a start:


Write $K = \overline{\mathbb{Q}\_{\ell}}$. One already sees the problems with what you are asking for $X\_0 = Spec \mathbb{F}\_q$. Then the category of constructible $K$-sheaves on $X\_0$ is equivalent to the category of finite dimensional $K$-representations of $Gal(\mathbb{F}/\mathbb{F}\_q)$, the absolute Galois group of $\mathbb{F}\_q$. (The absolute Galois group is generated topologically by Frobenius, and so this is the same as giving a finite dimensional $K$-vector space together with an endomorphism.) [See BBD 5.1.11 for a statement, I think this is explained in Milne, but don't have it at the moment.]


Now, a pure sheaf on $X\_0$ is pure of weight $i$ if all the eigenvalues of Frobenius are algebraic integers all of whose complex conjugates over $\mathbb{C}$ have the same absolute value $q^{i/2}$.


Note that here we already see that over $X\_0$ a pure sheaf does not need to be semi-simple. Indeed, there is no reason why Frobenius should act semi-simply. (This is one example of what de Cataldo and Migliorini are talking about in Remark 3.1.9 after the Theorem 3.1.8.) I think it is part of the standard conjectures that Frobenius acts semi-simply on the $\ell$-adic cohomology of smooth projective varieties, which as I understand it, is still not known.


I don't know what you mean when you ask:



> 
> Does the splitting of part 1 (only!) really requires $\overline{\mathbb{Q}\_{\ell}}$-coefficients?
> 
> 
> 


As to your main question, I think that the above example shows that this is too much to hope. Without working over $X\_0$ one cannot define what it means to be mixed, and without going to $X\_0$ one can't expect the same ext vanishing.


I recently discovered your work on "weight structures" and found it very interesting. I guess you are asking the above, because you would like to argue that one gets a weight structure in the setting of $\overline{\mathbb{Q}\_{\ell}}$-sheaves.


There is one setting where I think that one really does get a weight structure. This is in the (at least formally) very similar world of "mixed Hodge modules" on complex varieties. There one has the desired ext vanishing from the outset.


