Equivariant cohomology, syzygies and orbit structure

Let X X be a “nice” space with an action of a torus T T . We consider the Atiyah–Bredon sequence of equivariant cohomology modules arising from the filtration of X X by orbit dimension. We show that a front piece of this sequence is exact if and only if the H ∗ ( B T ) H^{*}(BT) -module H T ∗ ( X ) H_T^{*}(X) is a certain syzygy. Moreover, we express the cohomology of that sequence as an E x t \mathrm {Ext} module involving a suitably defined equivariant homology of X X . One consequence is that the GKM method for computing equivariant cohomology applies to a Poincaré duality space if and only if the equivariant Poincaré pairing is perfect.