Pure and Applied Mathematics Quarterly
Volume 11 (2015)
Bounding Betti numbers of monomial ideals in the exterior algebra
Pages: 267 – 281
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 $I$.
exterior algebra, monomial ideals, Betti numbers
2010 Mathematics Subject Classification
13A02, 15A75, 18G10