This paper studies the homology of finite modules over the exterior algebra
E
E
of a vector space
V
V
. To such a module
M
M
we associate an algebraic set
V
E
(
M
)
⊆
V
V_E(M)\subseteq V
, consisting of those
v
∈
V
v\in V
that have a non-minimal annihilator in
M
M
. A cohomological description of its defining ideal leads, among other things, to complementary expressions for its dimension, linked by a ‘depth formula’. Explicit results are obtained for
M
=
E
/
J
M=E/J
, when
J
J
is generated by products of elements of a basis
e
1
,
…
,
e
n
e_1,\dots ,e_n
of
V
V
. A (infinite) minimal free resolution of
E
/
J
E/J
is constructed from a (finite) minimal resolution of
S
/
I
S/I
, where
I
I
is the squarefree monomial ideal generated by ‘the same’ products of the variables in the polynomial ring
S
=
K
[
x
1
,
…
,
x
n
]
S=K[x_1,\dots ,x_n]
. It is proved that
V
E
(
E
/
J
)
V_E(E/J)
is the union of the coordinate subspaces of
V
V
, spanned by subsets of
{
e
1
,
…
,
e
n
}
\{\,e_1,\dots ,e_n\,\}
determined by the Betti numbers of
S
/
I
S/I
over
S
S
.