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

Abstract

For a given clutter $\\mathcal{C}$, let $I:=I ( \\bar{\\mathcal{C}} )$ be the\ncircuit ideal in the polynomial ring $S$. In this paper, we show that the Betti\nnumbers of $I$ and $I + ( \\textbf{x}_F )$ are the same in their non-linear\nstrands, for some suitable $F \\in \\mathcal{C}$. Motivated by this result, we\nintroduce a class of clutters that we call chordal. This class, is a natural\nextension of the class of chordal graphs and has the nice property that the\ncircuit ideal associated to any member of this class has a linear resolution\nover any field. Finally we compare this class with all known families of\nclutters which generalize the notion of chordality, and show that our class\ncontains several important previously defined classes of chordal clutters. We\nalso show that in comparison with others, this class is possibly the best\napproximation to the class of $d$-uniform clutters with linear resolution over\nany field.\n