Luca Amata, Marilena Crupi: Bounds for the Betti numbers of graded modules with given Hilbert function in an exterior algebra via lexicographic modules, 237-253

Abstract:

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 with all basis elements of the same degree. 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.

Key Words: Exterior algebra, Monomial ideal, Betti number, Hilbert function.

2010 Mathematics Subject Classification: Primary 13A02; Secondary 15A75, 18G10