  Using methods of homological algebra, we obtain an explicit crystal
isomorphism between two realizations of crystal bases of the lower part of the
quantized enveloping algebra of (almost all) finite dimensional simply-laced
Lie algebras. The first realization we consider is a geometric construction in
terms of irreducible components of certain quiver varieties established by
Kashiwara and Saito. The second is a realization in terms of isomorphism
classes of quiver representations obtained by Reineke using Ringel's Hall
algebra approach to quantum groups. We show that these two constructions are
closely related by studying sufficiently generic representations of the
preprojective algebra.
