Bounds for the Betti numbers of graded modules with given Hilbert function in an exterior algebra via lexicographic modules

Let $K$ be a field, $V$ a finite dimensional $K$-vector space, $E$ the exterior algebra of $V$, and $F$ a finitely generated graded free $E$-module. We prove that given any graded submodule $M$ of $F$, there exists a unique lexicographic submodule $L$ of $F$ such that $H_{F/L}=H_{F/M}$. As a consequence, we are able to describe the possible Hilbert functions of graded $E$-modules of the type $F/M$. Finally, we state that the lexicographic submodules of $F$ give the maximal Betti numbers among all the graded submodules of $F$ with the same Hilbert function.