  We show that the minimal free resolution of a general semi-stable sheaf $U$
on $\mathbb{P}^2$ contains a subcomplex that determines an extremal ray of the
cone of effective divisors of its moduli space. We provide evidence that this
is part of a general phenomenon in which minimal free resolutions, for distinct
Betti tables, contain subcomplexes depending on wall-crossing. From this
viewpoint, we provide new computations of the movable cones and Mori
decompositions of some moduli spaces of sheaves using syzygies.
