# zbMATH — the first resource for mathematics

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
##### Software:
Chordality; Chordal clutters
Full Text:
##### References:
 [1] A Program for Detecting Chordality. Available at: www.iasbs.ac.ir/ yazdan/chordality.html [2] A Program for computing numerical data of chordal clutters. Available at: www.iasbs.ac.ir/ yazdan/numericaldata.html [3] Adiprasito, KA; Nevo, E; Samper, JA, Higher chordality $${\rm I}$$: from graphs to complexes, Proc. Am. Math. Soc., 144, 3317-3329, (2016) · Zbl 1336.05149 [4] Bigdeli, M., Yazdan Pour, A.A., Zaare-Nahandi, R.: Stability of Betti numbers under reduction processes: towards chordality of clutters. J. Comb. Theory Ser. A 145, 129-149 (2017) · Zbl 1355.05285 [5] Bruns, W., Herzog, J.: Cohen-Macaulay Rings, Revised edn. Cambridge University Press, Cambridge (1996) · Zbl 0788.13005 [6] Conca, A; Herzog, J; Hibi, T, Rigid resolutions and big Betti numbers, Comment. Math. Helv., 79, 826-839, (2004) · Zbl 1080.13008 [7] Connon, E; Faridi, S, Chorded complexes and a necessary condition for a monomial ideal to have a linear resolution, J. Comb. Theory Ser. A, 120, 1714-1731, (2013) · Zbl 1314.05240 [8] Dirac, GA, On rigid circuit graphs, Abh. Math. Semin. Univ. Hambg., 38, 71-76, (1961) · Zbl 0098.14703 [9] Emtander, E, A class of hypergraphs that generalizes chordal graphs, Math. Scand., 106, 50-66, (2010) · Zbl 1183.05053 [10] Fröberg, R.: On Stanley-Reisner rings. In: Topics in Algebra, vol. 26, pp. 57-70. Polish Scientific Publishers (PWN), Warsaw (1990) · Zbl 0741.13006 [11] Herzog, J., Hibi, T.: Monomial Ideals, in GTM 260. Springer, London (2010) [12] Morales, M., Yazdan Pour, A.A., Zaare-Nahandi, R.: Regularity and free resolution of ideals which are minimal to $$d$$-linearity. Math. Scand. 118(2), 161-182 (2016) · Zbl 1350.13013 [13] Morales, M., Nasrollah Nejad, A., Yazdan Pour, A.A., Zaare-Nahandi, R.: Monomial ideals with $$3$$-linear resolutions. Ann. Fac. Sci. Toulouse Sér 6 23(4), 877-891 (2014) · Zbl 1304.13027 [14] Murai, S, Hilbert functions of $$d$$-regular ideals, J. Algebra, 317, 658-690, (2007) · Zbl 1144.13006 [15] Nikseresht, A., Zaare-Nahandi, R.: On generalizations of cycles and chordality to hypergraphs, preprint (2016). arXiv:1601.03207 · Zbl 1387.13046 [16] Woodroofe, R, Chordal and sequentially Cohen-Macaulay clutters, Electron. J. Comb., 18, 208-220, (2011) · Zbl 1236.05213
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.