Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
maximal depth; cycle graph; line graph; whisker graph; transversal polymatroidal ideal; power of edge ideal
maximal depth; cycle graph; line graph; whisker graph; transversal polymatroidal ideal; power of edge ideal
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.
References:
[1] Brodmann, M.: Asymptotic stability of Ass$(M/I^n M)$. Proc. Am. Math. Soc. 74 (1979), 16-18. DOI 10.2307/2042097 | MR 0521865 | Zbl A0395.13008
[2] Bruns, W., Herzog, J.: Cohen-Macaulay Rings. Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, Cambridge (1998). DOI 10.1017/cbo9780511608681 | MR 1251956 | Zbl 0909.13005
[3] Faridi, S.: Simplicial trees are sequentially Cohen-Macaulay. J. Pure Appl. Algebra 190 (2004), 121-136. DOI 10.1016/j.jpaa.2003.11.014 | MR 2043324 | Zbl 1045.05029
[4] Francisco, C. A., Hà, H. T.: Whiskers and sequentially Cohen-Macaulay graphs. J. Comb. Theory, Ser. A 115 (2008), 304-316. DOI 10.1016/j.jcta.2007.06.004 | MR 2382518 | Zbl 1142.13021
[5] Frühbis-Krüger, A., Terai, N.: Bounds for the regularity of monomial ideals. Mathematiche, Suppl. 53 (1998), 83-97. MR 1696019 | Zbl 0951.13017
[6] Herzog, J., Hibi, T.: Monomial Ideals. Graduate Texts in Mathematics 260, Springer, London (2011). DOI 10.1007/978-0-85729-106-6 | MR 2724673 | Zbl 1206.13001
[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. DOI 10.1007/s10801-012-0367-z | MR 3011344 | Zbl 1258.13014
[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. DOI 10.1090/tran/7662 | MR 3988608 | Zbl 1420.13023
[10] Martínez-Bernal, J., Morey, S., Villarreal, R. H.: Associated primes of powers of edge ideals. Collect. Math. 63 (2012), 361-374. DOI 10.1007/s13348-011-0045-9 | MR 2957976 | Zbl 1360.13027
[11] Miller, E., Sturmfels, B., Yanagawa, K.: Generic and cogeneric monomial ideals. J. Symb. Comput. 29 (2000), 691-708. DOI 10.1006/jsco.1999.0290 | MR 1769661 | Zbl 0955.13008
[12] Rahimi, A.: Maximal depth property of finitely generated modules. J. Algebra Appl. 17 (2018), Article ID 1850202, 12 pages. DOI 10.1142/S021949881850202X | MR 3879078 | Zbl 1409.13023
[13] Villarreal, R. H.: Monomial Algebras. Monographs and Research Notes in Mathematics, CRC Press, Boca Raton (2015). DOI 10.1201/b18224 | MR 3362802 | Zbl 1325.13004
Search
Partner of
