Derived categories of equivariant coherent sheaves

Many interesting varieties arise as a quotient orbifold $[X/G ]$ of a variety $X$ by an action of a finite group $G$. We are interested in the situation where there exists a resolution $\tau: Y \rightarrow X/G$ inducing an exact equivalence of derived categories $D^b(Y) \xrightarrow{\sim} D^b_G(X) \simeq D^b([X/G])$, where $D_G^b(X) := D^b(\mathfrak{Coh_G}X)$ is the bounded derived category of $G$-equivariant coherent sheaves on $X$. Any triangulated equivalence of those derived categories is necessarily a (generalised) Fourier-Mukai equivalence. One goal of this project is to render the above setup constructive when such resolutions $Y$ are known to exist.