# Which statement do people usually call the Decomposition Theorem, and what is the precise reference for it?

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
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
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
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