Let [Y] be an orbifold and Y the associated (singular) variety, also called the coarse moduli space of [Y]. The orbifold cohomology ring H^*_orb([Y]) of [Y] takes the local group actions, which are encoded in [Y], and the singularities of Y into account. Moreover it contains as a subring the singular cohomology of Y.

      Now, let r :Z ---> Y be a crepant resolution whose existence is assumed by hypothesis. In general, the cohomology rings H^*(Z) and H^*_orb([Y]) are not isomorphic. The cohomological crepant resolution conjecture (by Y. Ruan) states that the difference between the two rings can be expressed in terms of some Gromov-Witten invariants of Z of rational curves which are contracted by the resolution morphism r : Z ---> Y.

      We study this conjecture in the case where Y has transversal A_n singularities and [Y] is the associated orbifold. We compute both the orbifold cohomology H^*_orb([Y]) and the Gromov-Witten invariants of Z. Finally we prove the conjecture in the case A₁ and A₂ , giving in both cases an explicit isomorphism between the orbifold cohomology of [Y] and the quantum corrected cohomology ring of Z.