  Maximal green sequences were introduced as combinatorical counterpart for
Donaldson-Thomas invariants for 2-acyclic quivers with potential by B. Keller.
We take the categorical notion and introduce maximal green sequences for hearts
of bounded t-structures of triangulated categories that can be tilted
indefinitely. We study the case where the heart is the category of modules over
the preprojective algebra of a quiver without loops. The combinatorical
counterpart of maximal green sequences for Dynkin quivers are maximal chains in
the Hasse quiver of basic support \tau -tilting modules.
  We show that a quiver has a maximal green sequence if and only if it is of
Dynkin type. More generally, we study module categories for finite- dimensional
algebras with finitely many bricks.
