  Given $\mathfrak{F}$ a coherent sheaf on a Noetherian integral algebraic
stack $\mathfrak{P}$, we give two constructions of stacks
$\widetilde{\mathfrak{P}}$, equipped with birational morphisms
$p:\widetilde{\mathfrak{P}}\to \mathfrak{P}$ such that $p^*\mathfrak{F}$ is
simpler: in the Rossi construction, the torsion free part of $p^*\mathfrak{F}$
is locally free; in the Hu--Li diagonalization construction, $p^*\mathfrak{F}$
is a union of locally free sheaves. We use these constructions to define
reduced Gromov--Witten invariants of a large class of GIT quotients in all
genera.
