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# Pure and Applied Mathematics Quarterly

## Volume 11 (2015)

### Number 2

### Bounding Betti numbers of monomial ideals in the exterior algebra

Pages: 267 – 281

DOI: https://dx.doi.org/10.4310/PAMQ.2015.v11.n2.a4

#### Authors

#### Abstract

Let $K$ be a field, $V$ a $K$-vector space with basis $e_1, \dotsc , e_n$, 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 $I$.

#### Keywords

exterior algebra, monomial ideals, Betti numbers

#### 2010 Mathematics Subject Classification

13A02, 15A75, 18G10

Published 24 August 2016