Squarefree monomial ideals with maximal depth. (English) Zbl 07285983

Summary: Let \((R,\mathfrak{m})\) be a Noetherian local ring and \(M\) a finitely generated \(R\)-module. We say \(M\) has maximal depth if there is an associated prime \(\mathfrak{p}\) of \(M\) such that depth \(M=\dim R/\mathfrak{p}\). In this paper we study squarefree monomial ideals which have maximal depth. Edge ideals of cycle graphs, transversal polymatroidal ideals and high powers of connected bipartite graphs with this property are classified.


13C15 Dimension theory, depth, related commutative rings (catenary, etc.)
05E40 Combinatorial aspects of commutative algebra
Full Text: DOI arXiv


[1] Brodmann, M., Asymptotic stability of Ass \((M/I^n M)\), Proc. Am. Math. Soc. 74 (1979), 16-18
[2] Bruns, W.; Herzog, J., Cohen-Macaulay Rings, Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, Cambridge (1998)
[3] Faridi, S., Simplicial trees are sequentially Cohen-Macaulay, J. Pure Appl. Algebra 190 (2004), 121-136
[4] Francisco, C. A.; Hà, H. T., Whiskers and sequentially Cohen-Macaulay graphs, J. Comb. Theory, Ser. A 115 (2008), 304-316
[5] Frühbis-Krüger, A.; Terai, N., Bounds for the regularity of monomial ideals, Mathematiche, Suppl. 53 (1998), 83-97
[6] Herzog, J.; Hibi, T., Monomial Ideals, Graduate Texts in Mathematics 260, Springer, London (2011)
[7] Herzog, J.; Rauf, A.; Vladoiu, M., The stable set of associated prime ideals of a polymatroidal ideal, J. Algebr. Comb. 37 (2013), 289-312
[8] Jacques, S., Betti Numbers of Graph Ideals: Ph.D. Thesis, University of Sheffield, Sheffield (2004), Available at https://arxiv.org/abs/math/0410107\kern0pt
[9] Lam, H. M.; Trung, N. V., Associated primes of powers of edge ideals and ear decompositions of graphs, Trans. Am. Math. Soc. 372 (2019), 3211-3236
[10] Martínez-Bernal, J.; Morey, S.; Villarreal, R. H., Associated primes of powers of edge ideals, Collect. Math. 63 (2012), 361-374
[11] Miller, E.; Sturmfels, B.; Yanagawa, K., Generic and cogeneric monomial ideals, J. Symb. Comput. 29 (2000), 691-708
[12] Rahimi, A., Maximal depth property of finitely generated modules, J. Algebra Appl. 17 (2018), Article ID 1850202, 12 pages
[13] Villarreal, R. H., Monomial Algebras, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton (2015)
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.