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