Let T be a torus. We start by reviewing the theory of syzygies in equivariant cohomology. Among other things, it provides a characterization of those T-manifolds for which the equivariant Poincaré pairing is perfect and for which HT*(X) can be computed via the "GKM method". We also discuss extensions to actions of compact connected Lie groups.

Then we present a necessary and sufficient criterion for HT*(X) to be a certain syzygy as module over H^*(BT). It turns out that, possibly after blowing up the non-free part of the action, this only depends on the orbit space X/T together with its stratification by orbit type. Our criterion unifies and generalizes results of many authors about the freeness and torsion-freeness of equivariant cohomology for various classes of T-manifolds.