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Which statement is usually called the Decomposition Theorem (for perverse sheaves)? Is this (roughly): a proper pushforward of an intersection complex could be decomposed into a direct sum of (shifted) intersection complexes (or is it something more general, or possibly something less general)? Which theorems of [BBD] should one combine to get the 'usual' formulation of the decomposition theorem?

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I would understand the decomposition theorem to be precisely the result that you state in your second sentence. –  Emerton Feb 2 '11 at 1:57
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There is an excellent survey article in a recent issue of BAMS that might help, here's the PDF link: ams.org/journals/bull/2009-46-04/S0273-0979-09-01260-9/… –  Dave Anderson Feb 2 '11 at 2:33
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Mikhail, it is basically what you said, although there are some additional "purity" hypotheses, without which it can fail. The decomposition theorem is is given in [BBD, 6.2.5]. –  Donu Arapura Feb 2 '11 at 2:52
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In my comment above (as Donu Arapura's comment indicates), "precisely" is too strong a word. One form of the theorem says that pushing forward the IC complex upstairs (i.e. the one associated to the trivial local system) gives a direct sum of IC complexes of semisimple local systems downstairs; this is what I was thinking of when I made my comment. A more general version of the theorem (the one that Donu Arapura is referring to) describes the pushforward of IC sheaves attached to pure local systems upstairs. I think all this is discussed in the article that Dave Anderson links to. –  Emerton Feb 2 '11 at 9:30
    
Thanks; I know this paper. Yet it doesn't seem to contain a precise reference to BBD. –  Mikhail Bondarko Feb 2 '11 at 9:49
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