  Virtual resolutions are homological representations of finitely generated
$\text{Pic}(X)$-graded modules over the Cox ring of a smooth projective toric
variety. In this paper, we identify two algebraic conditions that characterize
when a chain complex of graded free modules over the Cox ring is a virtual
resolution. We then turn our attention to the saturation of Fitting ideals by
the irrelevant ideal of the Cox ring and prove some results that mirror the
classical theory of Fitting ideals for Noetherian rings.
