Some Semi-Classical Noncommutative Resolutions of Kleinian Singularities

We construct a class of noncommutative crepant resolutions of any Kleinian singularity in the form of noncommutative algebras over its crepant partial resolutions. We argue that such resolutions are Morita equivalent to the canonical orbifold resolutions of the partial resolutions. Further, we introduce Quot schemes which may be interpreted as Hilbert schemes of points for these orbifolds, and show that they are Nakajima quiver varieties.