Let
be a field,
a finite dimensional
-vector space,
the exterior algebra of
, and
a finitely generated graded free
-module with all basis elements of the same degree. We prove that given any graded submodule
of
, there exists a unique lexicographic submodule
of
such that
. As a consequence, we are able to describe the possible Hilbert functions of graded
-modules of the type
. Finally, we state that the lexicographic submodules of
give the maximal Betti numbers among all the graded submodules of
with the same Hilbert function.