An explicit construction of the McKay correspondence for A-Hilb C3

For a finite Abelian subgroup A⊂SL(3,C), let Y=A-Hilb(C3) denote the scheme parametrising A-clusters in C3. Ito and Nakajima proved that the tautological line bundles (indexed by the irreducible representations of A) form a basis of the K-theory of Y. We establish the relations between these bundles in the Picard group of Y and hence, following a recipe introduced by Reid, construct an explicit basis of the integral cohomology of Y in one-to-one correspondence with the irreducible representations of A.