The classical McKay correspondence relates the representation theory of a finite group G in SL(2,C) to the geometry of a resolution of the correspondent Kleinian singularity.

Given a quasiprojective nonsingular complex variety M and a finite group of automorphisms G, we can extend, under suitable hypotheses, this construction to an equivalence between the G-equivariant derived category of M and the the derived category of a crepant resolution of M/G, obtained via G-Hilbert schemes.

Since these conditions are always satisfied in dimension ≤ 3, we prove along the way a conjecture by Nakamura.

In this talk, we describe the projective case following the paper "The McKay correspondence as an equivalence of Derived Categories" by Bridgeland, King, and Reid, and defining many of the objects above.