Morphisms between pure complexes of sheaves

2010-04-27T08:20:01Z

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?

Answer by Geordie Williamson for Morphisms between pure complexes of sheaves

2010-05-02T13:52:30Z

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.

Morphisms between pure complexes of sheaves - MathOverflow

Morphisms between pure complexes of sheaves

Answer by Geordie Williamson for Morphisms between pure complexes of sheaves