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