  Let X be a manifold with an action of a torus T such that all isotropy groups
are connected and satisfying some other mild hypotheses. We provide a necessary
and sufficient criterion for the T-equivariant cohomology of X with real
coefficients to be a certain syzygy as a module over the cohomology of 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.
