  Associated to each finite group $\Gamma$ in $SL_2(C)$ there is a family of
noncommutative algebras which deforms the coordinate ring of the Kleinian
singularity corresponding to that group. These algebras were defined by W.
Crawley-Boevey and M. Holland, who also suggested a conjectural correspondence
between the set of isomorphism classes of rank one projective modules over
these algebras and associated Nakajima quiver varieties. In \cite{BGK},
V.Baranovski, V.Ginzburg and A.Kuznetsov proved the Crawley-Boevey-Holland
conjecture using the methods of noncommutative projective geometry. In this
paper we will state a refined ($G$-equivariant) version of this conjecture and,
in the case of cyclic groups, give a new construction of this correspondence
based on the notion of DG-model of a rank one projective module. This
construction leads to a completely explicit description of ideals of the
Crawley-Boevey-Holland algebras.
