Results on the normality of square-free monomial ideals and cover ideals under some graph operations
DOI:
https://doi.org/10.7146/math.scand.a-128963Abstract
In this paper, we introduce techniques for producing normal square-free monomial ideals from old such ideals. These techniques are then used to investigate the normality of cover ideals under some graph operations. Square-free monomial ideals that come out as linear combinations of two normal ideals are shown to be not necessarily normal; under such a case we investigate the integral closedness of all powers of these ideals.
References
Al-Ayyoub, I., Nasernejad, M., and Roberts, L., Normality of Monomial Ideals under some Operations with Application to Cover Ideals, Results Math. 74 (2019), no. 4, Paper No. 140, 26 pp. https://doi.org/10.1007/s00025-019-1068-x.
Bondy, J. A., and Murty, U. S. R., Graph theory. Graduate Texts in Mathematics, 244. Springer, New York, 2008. xii+651 pp. https://doi.org/10.1007/978-1-84628-970-5
Bruns, W., and Horn, M., Normaliz 3.8.4 and various earlier versions 3.0.0 and later with different authors. Documentation included with the program Normaliz: Algorithms for rational cones and affine monoids.
Available at https://www.normaliz.uni-osnabrueck.de/
Escobar, C., Villarreal, R. H., and Yoshino, Y., Torsion-freeness and normality of blowup rings of monomial ideals, Commutative Algebra, 69–84, Lect. Notes Pure Appl. Math. 244, Chapman & Hall/CRC, Boca Raton, FL, 2006. https://doi.org/10.1201/9781420028324.ch7
Faridi, S., Monomial Ideals via Square-free Monomial Ideals, Commutative Algebra, 85–114, Lecture Notes in Pure and Applied Math. 244, Chapman & Hall/CRC, Boca Raton, FL, 2006. https://doi.org/10.1201/9781420028324.ch8
Herzog, J., and Hibi, T., Monomial Ideals, Graduate Texts in Mathematics 260 Springer-Verlag London, Ltd., London, 2011. https://doi.org/10.1007/978-0-85729-106-6
Kaiser, T., Stehlík, M., and Škrekovski, R., Replication in critical graphs and the persistence of monomial ideals, J. Combin. Theory Ser. A 123 (2014), 239–251. https://doi.org/10.1016/j.jcta.2013.12.005
Khashyarmanesh K., and Nasernejad, M., On the minimal path cover sets of union of certain graphs, Quaest. Math. 43 (2020), no. 7, 881–901. https://doi.org/10.2989/16073606.2019.1586782
Martínez-Bernal, J., Morey, S., and Villarreal, R. H., Associated primes of powers of edge ideals, Collect. Math. 63 (2012), no. 3, 361–374. https://doi.org/10.1007/s13348-011-0045-9
Nasernejad, M., Asymptotic behaviour of associated primes of monomial ideals with combinatorial applications, J. Algebra Relat. Top. 2 (2014) 15–25.
Nasernejad, M., Persistence property for some classes of monomial ideals of a polynomial ring, J. Algebra Appl. 16, (2017), no. 6, 1750105, 17 pp. https://doi.org/10.1142/S0219498817501055
bibitem N3Nasernejad, M., and Khashyarmanesh, K., On the Alexander dual of the path ideals of rooted and unrooted trees, Comm. Algebra 45 (2017), no. 5, 1853–1864. https://doi.org/10.1080/00927872.2016.1226855
Nasernejad, M., and Khashyarmanesh, K., Sets defining minimal vertex covers of uniform hypergraphs, Ars Combin. 147 (2019) 135–142.
Nasernejad, M., Khashyarmanesh, K., and Al-Ayyoub, I., Associated primes of powers of cover ideals under graph operations, Comm. Algebra, 47 (2019), no. 5, 1985–1996. https://doi.org/10.1080/00927872.2018.1527920
Rajaee, S., Nasernejad, M., and Al-Ayyoub, I., Superficial ideals for monomial ideals, J. Algebra Appl. 16 (2018), no. 6, 1850102, 28 pp. https://doi.org/10.1142/S0219498818501025
Restuccia. G., and Villarreal, R. H., On the normality of monomial ideals of mixed products, Comm. Algebra 29 (2001), no. 8, 3571–3580. https://doi.org/10.1081/AGB-100105039
Sharifan, L., Nasernejad, M., and Khashyarmanesh, K., Minimal path cover sets and monomial ideals, J. Algebra Appl. 14 (2015), no. 2, 1550014, 14 pp. https://doi.org/10.1142/S0219498815500140
Simis, A., Vasconcelos, W., and Villarreal, R. H., On the ideal theory of graphs, J. Algebra 167 (1994), no. 2, 389–416. https://doi.org/10.1006/jabr.1994.1192
Villarreal, R. H., Monomial algebras, Second edition. Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, FL, 2015.
Villarreal, R. H., Normality of subrings generated by square-free monomials, J. Pure Appl. Algebra 113 (1996), no. 1, 91–106. https://doi.org/10.1016/0022-4049(95)00145-X
Villarreal, R. H., Rees cones and monomial rings of matroids, Linear Algebra Appl. 428 (2008), no. 11–12, 2933–2940. https://doi.org/10.1016/j.laa.2008.01.035
Villarreal, R. H., Rees algebras and polyhedral cones of ideals of vertex covers of perfect graphs, J. Algebraic Combin. 27 (2008), no. 6, 293–305. https://doi.org/10.1007/s10801-007-0088-x
