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Simplicial orders and chordality. (English) Zbl 1436.13028
Summary: Chordal clutters in the sense of M. Bigdeli et al. [J. Comb. Theory, Ser. A 145, 129–149 (2017; Zbl 1355.05285)] and M. Morales et al. [Ann. Fac. Sci. Toulouse, Math. (6) 23, No. 4, 877–891 (2014; Zbl 1304.13027)] 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 \(\lambda \)-sequence of the chordal clutter. All possible \(\lambda \)-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 \(\lambda \)-sequence of a chordal clutter, we determine other numerical invariants of the circuit ideal, such as the \(\mathbf h \)-vector and the Betti numbers.

MSC:
13D02 Syzygies, resolutions, complexes and commutative rings
13P20 Computational homological algebra
05E45 Combinatorial aspects of simplicial complexes
05C65 Hypergraphs
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