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Using methods of homological algebra, we obtain an explicit crystalisomorphism between two realizations of crystal bases of the lower part of thequantized enveloping algebra of (almost all) finite dimensional simply-lacedLie algebras. The first realization we consider is a geometric construction interms of irreducible components of certain quiver varieties established byKashiwara and Saito. The second is a realization in terms of isomorphismclasses of quiver representations obtained by Reineke using Ringel's Hallalgebra approach to quantum groups. We show that these two constructions areclosely related by studying sufficiently generic representations of thepreprojective algebra.

Homological description of crystal structures on Lusztig's quiver varieties