Abstract
Let \(G\) be a graph, and let \(I\) be the edge ideal of \(G\). Our main results in this article provide lower bounds for the depth of the first three powers of \(I\) in terms of the diameter of \(G\). More precisely, we show that \(\mathrm{{depth}}\,R/I^t \ge \left\lceil {\frac{d-4t+5}{3}} \right\rceil +p-1\), where \(d\) is the diameter of \(G\) and \(p\) is the number of connected components of \(G\) and \(1 \le t \le 3\). For general powers of edge ideals we show that \(\mathrm{{depth}}\,R/I^t \ge p-t\). As an application of our results we obtain the corresponding lower bounds for the Stanley depth of the first three powers of \(I\).
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References
Bandari, S., Herzog, J., Hibi, T.: Monomial ideals whose depth function has any given number of strict local maxima. Ark. Mat. 52, 11–19 (2014)
Brodmann, M.: The asymptotic nature of the analytic spread. Math. Proc. Camb. Philos. Soc. 86, 35–39 (1979)
Bruns, W., Herzog, J.: Cohen–Macaulay Rings, Revised Edition. Cambridge University Press, Cambridge (1997)
Bruns, W., Krattenthaler, C., Uliczka, J.: Stanley decompositions and Hilbert depth in the Koszul complex. J. Commut. Algebra 2, 327–357 (2010)
Burch, L.: Codimension and analytic spread. Proc. Camb. Philos. Soc. 72, 369–373 (1972)
Chen, J., Morey, S., Sung, A.: The stable set of associated primes of the ideal of a graph. Rocky Mountain J. Math. 32, 71–89 (2002)
Dao, H., Huneke, C., Schweig, J.: Bounds on the regularity and projective dimension of ideals associated to graphs. J. Algebraic Combin. 38, 37–55 (2013)
Dao, H., Schweig, J.: Projective dimension, graph domination parameters, and independence complex homology. J. Combin. Theory Ser. A 120, 453–469 (2013)
Dochtermann, A., Engström, A.: Algebraic properties of edge ideals via combinatorial topology. Electron. J. Combin. 16 (2009) (Special volume in honor of Anders Björner, Research Paper 2, 24 pp)
Duval, A.M., Goeckner, B., Klivans, C.J., Martin, J.L.: A non-partitionable Cohen–Macaulay simplicial complex. arXiv:1504.04279 [math.CO]
Eisenbud, D., Huneke, C.: Cohen–Macaulay Rees algebras and their specializations. J. Algebra 81, 202–224 (1983)
Eisenbud, D., Mustaţǎ, M., Stillman, M.: Cohomology on toric varieties and local cohomology with monomial supports. J. Symb. Comput 29, 583–600 (2000)
Engström, A., Noren, P.: Cellular resolutions of powers of monomial ideals. arXiv:1212.2146 [math.AC]
Faltings, G.: Uber lokale Kohomologiegruppen hoher Ordnung. J. fur Reine Angewandte Math 313, 43–51 (1980)
Francisco, C., Ha, H.T., Van Tuyl, A.: Associated primes of monomial ideals and odd holes in graphs. J. Algebraic Comb. 32, 287–301 (2010)
Gitler, I., Valencia, C.: Bounds for invariants of edge rings. Commun. Algebra 33, 1603–1616 (2005)
Grayson, D.R., Stillman, M.E.: Macaulay2, a software system for research in algebraic geometry. http://www.math.uiuc.edu/Macaulay2/
Hà, H.T., Morey, S.: Embedded associated primes of powers of square-free monomial ideals. J. Pure Appl. Algebra 214, 301–308 (2010)
Hartshorne, R.: Cohomological dimension of algebraic varieties. Ann. Math. 88, 403–450 (1968)
Herzog, J.: A survey on Stanley depth, Monomial ideals, computations and applications, 3–45, Lecture Notes in Math, 2083. Springer, Heidelberg (2013)
Herzog, J., Hibi, T.: The depth of powers of an ideal. J. Algebra 291, 534–550 (2005)
Hà, H.T., Trung, N., Trung, T.: Depth and regularity of powers of sums of ideals. arXiv:1501.06038v1 [math.AC]
Kaiser, T., Stehlík, M., Škrekovski, R.: Replication in critical graphs and the persistence of monomial ideals. J. Comb. Theory Ser. A 123, 239–251 (2014)
Kummini, M.: Regularity, depth and arithmetic rank of bipartite edge ideals. J. Algebraic Comb. 30, 429–445 (2009)
Lyubeznik, G.: On some local cohomology modules. Adv. Math. 213, 621–643 (2007)
Matsumura, H.: Commutative Ring Theory, Cambridge Studies in Advanced Math, vol. 8. Cambridge University Press, Cambridge (1986)
Morey, S.: Depths of powers of the edge ideal of a tree. Commun. Algebra 38, 4042–4055 (2010)
Ogus, A.: Local cohomological dimension of algebraic varieties. Ann. Math. 98, 999–1013 (1976)
Peskine, C., Szpiro, L.: Dimension projective finie et cohomologie locale. Inst. Hautes Etudes Sci. Publ. Math. 42, 47–119 (1973)
Pournaki, M.R., Seyed Fakhari, S.A., Yassemi, S.: Stanley depth of powers of the edge ideal of a forest. Proc. Am. Math. Soc. 141, 3327–3336 (2013)
Rauf, A.: Depth and Stanley depth of multigraded modules. Commun. Algebra 38, 773–784 (2010)
Simis, A., Vasconcelos, W.V., Villarreal, R.H.: On the ideal theory of graphs. J. Algebra 167, 389–416 (1994)
Singh, A., Walther, U.: Local cohomology and pure morphisms. Ill. J. Math. 51, 287–298 (2007)
Stanley, R.P.: Linear diophantine equations and local cohomology. Invent. Math. 68, 175–193 (1982)
Terai, N.: Alexander duality theorem and Stanley-Reisner rings. Sürikaisekikenkyüsho Kökyüroku (1999), 174–184. Free resolutions of coordinate rings of projective varieties and related topics (Japanese) (Kyoto, 1998)
Villarreal, R.H.: Monomial Algebras, Monographs and Textbooks in Pure and Applied Mathematics, vol. 238. Marcel Dekker Inc, New York (2001)
Acknowledgments
The authors would like to thank Nate Dean for helpful conversations regarding graph theory and the anonymous referees for useful comments and suggestions. The first author would also like to thank the Department of Mathematics at Texas State University for its hospitality while some of the work was completed.
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The first author was partially supported by a grant from the Simons Foundation, Grant #244930.
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Fouli, L., Morey, S. A lower bound for depths of powers of edge ideals. J Algebr Comb 42, 829–848 (2015). https://doi.org/10.1007/s10801-015-0604-3
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DOI: https://doi.org/10.1007/s10801-015-0604-3