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

For a given clutter CC, let I:=I(C¯) be the circuit ideal in the polynomial ring S. In this paper, we show that the Betti numbers of I   and I+(xF)I+(xF) are the same in their non-linear strands, for some suitable F∈CF∈C. Motivated by this result, we introduce a class of clutters that we call chordal. This class is a natural extension of the class of chordal graphs and has the nice property that the circuit ideal associated to the complement of any member of this class has a linear resolution over any field. Finally we compare this class with all known families of clutters which generalize the notion of chordality, and show that our class contains several important previously defined classes of chordal clutters.