There's an important piece of geometric knowledge usually quoted as Beilinson-Bernstein-Deligne.

Here's a refresher: by `IC` I mean the intersection complex, the one that is just `Q` for a smooth scheme but more complicated for others, by `IC_i` its version correctly extended (the `!*` notation) from a subscheme of `Y` (with bundle). 

Then it turns out that
for a morphism `f: X\to Y`, you can decompose, in the derived category, 

    f_*IC  =  direct sum of things of the form IC_i[n]. 

The special beauty of this decomposition theorem is in its examples. Here are some I think I know:

* For a **free action** of a group G on some X, you get the decomposition by representation of G.
* For a **resolution of singularities**, you get `f_*Q = Q \oplus F`, (F is on the exceptional divisor.)
* For a **smooth algebraic bundle** `f_*Q = sum of Q` (spectral sequence degenerates)

Did I make any mistake? What are other examples, especially the important special cases?