HILBERT FUNCTIONS OF GRADED MODULES OVER AN EXTERIOR ALGEBRA: AN ALGORITHMIC APPROACH

Abstract

Let $K$ be a field, $E$ the exterior algebra of a finite dimensional $K$-vector space, and $F$ a finitely generated graded free $E$-module with homogeneous basis $g_1, \ldots, g_r$ such that $\deg g_1 \le \deg g_2 \le \cdots \le \deg g_r$. Given the Hilbert function of a graded $E$--module of the type $F/M$, with $M$ graded submodule of $F$, the existence of the unique lexicographic submodule of $F$ with the same Hilbert function as $M$ is proved by a new algorithmic approach. Such an approach allows us to establish a criterion for determining if a sequence of nonnegative integers defines the Hilbert function of a quotient of a free $E$--module only via the combinatorial Kruskal--Katona's theorem.