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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theIC 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. $\endgroup$ – Emerton Feb 2 '11 at 9:30