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.