| We use stacky compactifications of minimal resolutions of singularities of type A_k (ALE spaces) and a theory of framed sheaves on projective stacks to give rigorous definitions of partition functions for supersymmetric gauge theories on ALE spaces. We show that in dimension 2 the moduli functor for framed sheaves on projective stacks is representable, and is represent by a scheme, for which it is possible to compute the obstruction to smoothness. For the case of the stacky compactifications of A_k ALE spaces, that scheme contains a dense open subvariety which is a moduli spaces for instantons on the ALE spaces with fixed Chern classes and holonomy at infinity. |