Bounding Betti numbers of monomial ideals in the exterior algebra

Marilena Crupi; Carmela Ferro'

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arXiv · 1602.08864

Bounding Betti numbers of monomial ideals in the exterior algebra

Marilena Crupi·Carmela Ferro'

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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 $I$.

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Marilena Crupi, Carmela Ferro'. 2016-02-29. Bounding Betti numbers of monomial ideals in the exterior algebra. https://arxiv.org/abs/1602.08864

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