# What is the purpose of section 3 of BBD?

I am not quite sure that this question is appropriate for Mathoverflow, yet I would be deeply grateful for any hint: what happens in section 3 of Beilinson A., Bernstein J., Deligne P., Faisceaux pervers//Asterisque 100, 1982, 5-171? Is there any statement that is important for the following sections of this treatise? I do not know French; yet this does not prevent me from understanding all the other sections.

Upd. I wonder: does there exist a 'plan' of BBD?

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I think this question is completely appropriate for Mathoverflow. Certainly research-level (or beyond :-), and typically a question for which help by some expert(e) who knows BBD inside-out can be invaluable. – Joël Apr 11 '13 at 20:24

I'm not an expert at BBD, but the introduction explains that chapter 3 is supplementary technical information that can be skipped. For example, if you want to work with $\mathbb{Z}$-coefficients, or the filtered derived category, you might find chapter 3 useful. As far as a "plan" of BBD is concerned, the introduction has a brief description of the contents of each chapter, but if you are looking for some kind of "tree" of lemmata leading to main results like the decomposition theorem, I haven't seen one.

I only found 3 references to chapter 3 in the rest of the book:

1. The proof of Proposition 2.1.23 uses section 3.2.

2. Part 2.2.19 gives a proof of Proposition 2.1.23 that does not use results of chapter 3.

3. The first page of chapter 4 mentions section 3.3 in an inessential way.

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Note that the filtered derived category is of practical use elsewhere: for example, Beilinson's "On the derived category of perverse sheaves", where he invokes a functor from the eponymous category to the usual derived category, constructed using the filtered derived category. It is this functor that is shown to be an equivalence. Granted the definition is pretty technical, but even just for the sake of this result it seems like it is a valid piece of the theory and not just architecture. – Ryan Reich Apr 12 '13 at 4:19
I know that the filtered derived category was used in succeeding papers. Yet I was not able to understand whether it was used in BBD. – Mikhail Bondarko Apr 12 '13 at 7:56