  Big polygon spaces are compact orientable manifolds with a torus action whose
equivariant cohomology can be torsion-free or reflexive without being free as a
module over $H^*(BT)$. We determine the exact syzygy order of the equivariant
cohomology of a big polygon space in terms of the length vector defining it.
The proof uses a refined characterization of syzygies in terms of certain
linearly independent elements in $H^2(BT)$ adapted to the isotropy groups
occurring in a given $T$-space.
