Moduli spaces of framed sheaves on compactified Kleinian singularities

Consider a Kleinian singularity ${\mathbb{C}}^2\mathrm{/}\mathrm{\Gamma }$, where $\mathrm{\Gamma }$ is a finite subgroup of $S{L}_2(\mathbb{C})$. In this paper, we construct moduli spaces of framed sheaves on a projective Deligne-Mumford stack compactifying the singularity, and we show that these moduli spaces are quasiprojective schemes.