Bounding Betti numbers of monomial ideals in the exterior algebra

TL;DR

This paper investigates bounds on Betti numbers of monomial ideals in exterior algebras, introducing colexsegment ideals and proving a lower bound for strongly stable ideals generated in one degree.

Contribution

It introduces colexsegment ideals in exterior algebra and proves a Betti number lower bound for strongly stable ideals generated in one degree.

Findings

Colexsegment ideals have minimal Betti numbers among strongly stable ideals.

Strongly stable ideals generated in one degree satisfy the colex lower bound.

Betti numbers of colexsegment ideals are less than or equal to those of the original ideals.

Abstract

Let $K$ be a field, $V$ a $K$-vector space with basis $e_1,\ldots,e_n$, and $E$ the exterior algebra of $V$. To a given monomial ideal $I\subsetneq E$ we associate a special monomial ideal $J$ with generators in the same degrees as those of $I$ and such that the number of the minimal monomial generators in each degree of $I$ and $J$ coincide. We call $J$ the colexsegment ideal associated to $I$. We prove that the class of strongly stable ideals in $E$ generated in one degree satisfies the colex lower bound, that is, the total Betti numbers of the colexsegment ideal associated to a strongly stable ideal $I\subsetneq E$ generated in one degree are smaller than or equal to those of $I$.