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

The syzygy order of big polygon spaces