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.

The index of a graded ideal measures the number of linear steps in the graded minimal free resolution of the ideal. In this paper we study the index of powers and squarefree powers of edge ideals. Our results indicate that the index as a function of the power of an edge ideal I is strictly increasing if I has linear relations. Examples show that this need not to be the case for monomial ideals generated in degree greater than two.

a b s t r a c tLet A(C) be the coordinate ring of a monomial curve C ⊆ A n corresponding to the numerical semigroup S minimally generated by a sequence a 0 , . . . , a n . In the literature, little is known about the Betti numbers of the corresponding associated graded ring gr m (A) with respect to the maximal ideal m of A = A(C). In this paper we characterize the numerical invariants of a minimal free resolution of gr m (A) in the case a 0 , . . . , a n is a generalized arithmetic sequence.

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.

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