Pure and Applied Mathematics Quarterly

Volume 11 (2015)

Number 2

Bounding Betti numbers of monomial ideals in the exterior algebra

Pages: 267 – 281

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

Authors

Marilena Crupi (Dipartimento di Scienze Matematiche e Informatiche, University of Messina, Italy)

Carmela Ferrò (Dipartimento di Scienze Matematiche e Informatiche, University of Messina, Italy)

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

Full Text (PDF format)