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.

Abstract. We call a simplicial complex algebraically rigid if its Stanley-Reisner ring admits no nontrivial infinitesimal deformations, and call it inseparable if it does not allow any deformation to other simplicial complexes. Algebraically rigid simplicial complexes are inseparable. In this paper we study inseparability and rigidity of Stanley-Reisner rings, and apply the general theory to letterplace ideals as well as to edge ideals of graphs. Classes of algebraically rigid simplicial complexes and graphs are identified.

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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