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.

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

Amata, Luca;Crupi, Marilena
2018

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. 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.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11570/3131750
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