For a given clutter C, 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 ) are the same in their non-linear strands, for some suitable F ∈ 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 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. We also show that in comparison with others, this class is possibly the best approximation to the class of d-uniform clutters with linear resolution over any field.2010 Mathematics Subject Classification. Primary 13D02, 13F55; Secondary 05E45, 05C65.

In this paper, we study Cstelnuovo-Mumford regularity of square-free monomial ideals generated in degree 3. We define some operations on the clutters associated to such ideals and prove that the regularity is conserved under these operations. We apply the operations to introduce some classes of ideals with linear resolutions and also show that any clutter corresponding to a triangulation of the sphere does not have linear resolution while any proper sub-clutter of it has a linear resolution.

Chordal clutters in the sense of [14] and [3] are defined via simplicial orders. Their circuit ideal has a linear resolution, independent of the characteristic of the base field. We show that any Betti sequence of an ideal with linear resolution appears as the Betti sequence of the circuit ideal of such a chordal clutter. Associated with any simplicial order is a sequence of integers which we call the λ-sequence of the chordal clutter. All possible λ-sequences are characterized. They are intimately related to the Hilbert function of a suitable standard graded K-algebra attached to the chordal clutter. By the λ-sequence of a chordal clutter we determine other numerical invariants of the circuit ideal, such as the h-vector and the Betti numbers.2010 Mathematics Subject Classification. Primary 13D02, 13P20; Secondary 05E45, 05C65.

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