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

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 homo- geneous basis g_1,...., g_r such that deg g_1= deg g_2= ...= deg g_r. Given the Hilbert function of a graded E-module of the type F=M, with M graded sub- module 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.