Stability of Betti numbers under reduction processes: towards chordality of clutters

摘要

For a given clutter $\mathcal{C}$, let $I:=I ( \bar{\mathcal{C}} )$ be thecircuit ideal in the polynomial ring $S$. In this paper, we show that the Bettinumbers of $I$ and $I + ( \textbf{x}_F )$ are the same in their non-linearstrands, for some suitable $F \in \mathcal{C}$. Motivated by this result, weintroduce a class of clutters that we call chordal. This class, is a naturalextension of the class of chordal graphs and has the nice property that thecircuit ideal associated to any member of this class has a linear resolutionover any field. Finally we compare this class with all known families ofclutters which generalize the notion of chordality, and show that our classcontains several important previously defined classes of chordal clutters. Wealso show that in comparison with others, this class is possibly the bestapproximation to the class of $d$-uniform clutters with linear resolution overany field.