# In what setting does one usually define mixed sheaves and weights for them?

In BBD mixed sheaves and weights for them were only defined for ($\overline{\mathbb{Q}_l}$-)sheaves over a variety $X_0$ defined over a finite field $F$. Weights start to behave better when one extends coefficients from $F$ to its algebraic closure i.e. passes from $X_0$ to $X$.

Now, BBD was published in 1982. Are any significant improvements and/or generalizations known in this field now? There is a paper by Huber and a book by Jannsen where mixed sheaves and weights for them are mentioned. Yet these authors were not able to generalize the results of BBD (in fact, an example of Jannsen shows that this is probably impossible). They also didn't extend scalars. So, are there any other papers on this subject?

Upd. There seems to be two basic ways to define weights (for sheaves) explicitly. The first method uses weights of Hodge structures. It seems that this method can work only for something like the category of mixed Hodge modules. Possibly, I will study these categories in the future. Yet at the moment I study motives, and it seems that 'motivic' people usually do not understand mixed Hodge modules (and so did not relate them with motives).

So, I am currently interested in the second method. It uses the eigenvalues of the Frobenius action. So, was anything interesting done using THIS approach after 1982?

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Dear Mikhail, If you are not sure about the role of eigenvalues of Frobenius and their relations to weights, and also the relations to Hodge theory, I suggest you go back before 1982 and read Deligne's ICM address, Theorie de Hodge I. Mixed Hodge modules are intimately related to motives. The idea of motives and mixed sheaves is that there is an underlying category of mixed motivic sheaves (here "mixed" means "not necessarily pure", i.e. not necessarily arising as the cohomology of a smooth proper family over space where the sheaf lives). –  Emerton Jul 9 '10 at 23:45
(Note: in the above, the sentence beginning "Mixed Hodge modules ... " was supposed to be the start of a new thought, and a new paragraph; but unfortunately comments don't seem to allow paragraphs. Anyway, continuing ... ) There are then realization functors to the various cohomology theories: $\ell$-adic realization takes mixed motivic sheaves to perverse sheaves as in BBD, and the Hodge theoretic realization takes values in mixed Hodge modules. A final remark: you might want to ask a more specific question then "was anything interesting done ...", since you are asking about a ... –  Emerton Jul 9 '10 at 23:49
... fundamental technique and topic in algebraic geometry, arithmetic geometry, and number theory. People use these tools all the time, and are always trying to improve them (by trying to prove everything that has been conjectured, e.g. the weight-monodromy conjecture). The problem of constructing the category of mixed motivic sheaves is a fundamental and difficult one, but much thought has been given to that too, and the philosophy it suggests is used frequently as a guide to the truth. –  Emerton Jul 9 '10 at 23:52
Dear Emerton, thank you! I am in the following situation: I know something about the motivic side of the story (that depends on several conjectures), and want to know more about sheaf-theoretic and non-conjectual things. –  Mikhail Bondarko Jul 10 '10 at 0:57
These are very good answers. I have nothing to add except perhaps to suggest also looking at Deligne's 1974 ICM talk "Poids dans la cohomologie des variétés algébriques". –  Donu Arapura Jul 10 '10 at 1:09

Admittedly, I'm almost completely ignorant about the $\ell$-adic setting. But, in case the following at least gets at the spirit of your question: in the de Rham setting, I found an old (1990s) preprint of Saito ("On the formalism of mixed sheaves," now TeXed up and available on the arxiv) useful in understanding the formal structure of (the system of categories of) mixed sheaves. The book by Peters and Steenbrink ("Mixed Hodge Structures") also has a nice section (14.1, "An axiomatic introduction") in the chapter (14) on mixed Hodge modules that explains the picture well.
I'm not sure if this is the kind of answer that you're looking for. This is an extension of Tom Nevin's answer. Saito's theory of mixed Hodge modules is modeled, to some extent, on BBD. The results are quite similar, but the constructions and proofs are entirely different (and unfortunately rather opaque). In a sense, this follows a general pattern. In the $\ell$-adic world, the weight filtration is determined naturally from the Galois action. In Hodge theory, one usually has to guess what it is based on analogy with it.
Perhaps I can say a few more words to make the formal structure of Saito's theory a little clearer. A mixed Hodge module consists a filtered perverse sheaf $(K,W)$ over $\mathbb{Q}$, and a bifiltered regular holonomic $D$-module $(M,W,F)$ such that $(M,W)$ corresponds to $(K,W)$ under Riemann-Hilbert. One then needs to impose some axioms in order to get good theory. The first is that over point, this datum defines a mixed Hodge structure. Before getting to the second, recall that one knows from BBD (and G I guess) that mixed perverse sheaves are stable under vanishing cycles. Saito takes this as an axiom. However, making that last statement precise takes over 100 pages.