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.