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

Authors : Luca AMATA, Marilena CRUPİ

Pages : 271-287

Doi:10.24330/ieja.663094

View : 13 | Download : 8

Publication Date : 2020-01-07

Article Type : Research Paper

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.
Keywords : Exterior algebra, Hilbert function, monomial submodule, lexicographic submodule