Associated primes of powers of edge ideals and ear decompositions of graphs

Lam, Ha; Trung, Ngo

doi:10.1090/tran/7662

Associated primes of powers of edge ideals and ear decompositions of graphs
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by Ha Minh Lam and Ngo Viet Trung PDF

Trans. Amer. Math. Soc. 372 (2019), 3211-3236 Request permission

Abstract:

In this paper, we give a complete description of the associated primes of every power of the edge ideal in terms of generalized ear decompositions of the graph. This result establishes a surprising relationship between two seemingly unrelated notions of commutative algebra and combinatorics. It covers all previous major results in this topic and has several interesting consequences.

References

Arindam Banerjee, The regularity of powers of edge ideals, J. Algebraic Combin. 41 (2015), no. 2, 303–321. MR 3306074, DOI 10.1007/s10801-014-0537-2
M. Brodmann, Asymptotic stability of $\textrm {Ass}(M/I^{n}M)$, Proc. Amer. Math. Soc. 74 (1979), no. 1, 16–18. MR 521865, DOI 10.1090/S0002-9939-1979-0521865-8
M. Brodmann, The asymptotic nature of the analytic spread, Math. Proc. Cambridge Philos. Soc. 86 (1979), no. 1, 35–39. MR 530808, DOI 10.1017/S030500410000061X
Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
Janet Chen, Susan Morey, and Anne Sung, The stable set of associated primes of the ideal of a graph, Rocky Mountain J. Math. 32 (2002), no. 1, 71–89. MR 1911348, DOI 10.1216/rmjm/1030539608
S. Dale Cutkosky, Jürgen Herzog, and Ngô Viêt Trung, Asymptotic behaviour of the Castelnuovo-Mumford regularity, Compositio Math. 118 (1999), no. 3, 243–261. MR 1711319, DOI 10.1023/A:1001559912258
N. T. Dung, N. T. T. Tam, H. L. Truong, and H. N. Yen, Critical paired dominating sets and irreducible decompositions of powers of edge ideals, Acta Math. Vietnam. (2018). https://doi.org/10.1007/s40306-018-0273-0
David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
David Eisenbud and Joe Harris, Powers of ideals and fibers of morphisms, Math. Res. Lett. 17 (2010), no. 2, 267–273. MR 2644374, DOI 10.4310/MRL.2010.v17.n2.a6
David Eisenbud and Bernd Ulrich, Notes on regularity stabilization, Proc. Amer. Math. Soc. 140 (2012), no. 4, 1221–1232. MR 2869107, DOI 10.1090/S0002-9939-2011-11270-X
Christopher A. Francisco, Huy Tài Hà, and Adam Van Tuyl, Colorings of hypergraphs, perfect graphs, and associated primes of powers of monomial ideals, J. Algebra 331 (2011), 224–242. MR 2774655, DOI 10.1016/j.jalgebra.2010.10.025
András Frank, Conservative weightings and ear-decompositions of graphs, Combinatorica 13 (1993), no. 1, 65–81. MR 1221177, DOI 10.1007/BF01202790
T. Gallai, Neuer Beweis eines Tutte’schen Satzes, Magyar Tud. Akad. Mat. Kutató Int. Közl. 8 (1963), 135–139 (German, with Russian summary). MR 166777
Huy Tài Hà and Susan Morey, Embedded associated primes of powers of square-free monomial ideals, J. Pure Appl. Algebra 214 (2010), no. 4, 301–308. MR 2558739, DOI 10.1016/j.jpaa.2009.05.002
Huy Tài Hà and Mengyao Sun, Squarefree monomial ideals that fail the persistence property and non-increasing depth, Acta Math. Vietnam. 40 (2015), no. 1, 125–137. MR 3331937, DOI 10.1007/s40306-014-0104-x
Jürgen Herzog and Takayuki Hibi, The depth of powers of an ideal, J. Algebra 291 (2005), no. 2, 534–550. MR 2163482, DOI 10.1016/j.jalgebra.2005.04.007
Jürgen Herzog and Takayuki Hibi, Monomial ideals, Graduate Texts in Mathematics, vol. 260, Springer-Verlag London, Ltd., London, 2011. MR 2724673, DOI 10.1007/978-0-85729-106-6
Jürgen Herzog and Takayuki Hibi, Bounding the socles of powers of squarefree monomial ideals, Commutative algebra and noncommutative algebraic geometry. Vol. II, Math. Sci. Res. Inst. Publ., vol. 68, Cambridge Univ. Press, New York, 2015, pp. 223–229. MR 3496867
Jürgen Herzog, Takayuki Hibi, and Ngô Viêt Trung, Symbolic powers of monomial ideals and vertex cover algebras, Adv. Math. 210 (2007), no. 1, 304–322. MR 2298826, DOI 10.1016/j.aim.2006.06.007
Jürgen Herzog and Ayesha Asloob Qureshi, Persistence and stability properties of powers of ideals, J. Pure Appl. Algebra 219 (2015), no. 3, 530–542. MR 3279372, DOI 10.1016/j.jpaa.2014.05.011
Ha Thi Thu Hien and Ha Minh Lam, Combinatorial characterizations of the saturation and the associated primes of the fourth power of edge ideals, Acta Math. Vietnam. 40 (2015), no. 3, 511–526. MR 3395652, DOI 10.1007/s40306-015-0140-1
Ha Thi Thu Hien, Ha Minh Lam, and Ngo Viet Trung, Saturation and associated primes of powers of edge ideals, J. Algebra 439 (2015), 225–244. MR 3373370, DOI 10.1016/j.jalgebra.2015.05.003
Lê Tuân Hoa, Stability of associated primes of monomial ideals, Vietnam J. Math. 34 (2006), no. 4, 473–487. MR 2286753
Vijay Kodiyalam, Asymptotic behaviour of Castelnuovo-Mumford regularity, Proc. Amer. Math. Soc. 128 (2000), no. 2, 407–411. MR 1621961, DOI 10.1090/S0002-9939-99-05020-0
L. Lovász, A note on factor-critical graphs, Studia Sci. Math. Hungar. 7 (1972), 279–280. MR 335371
José Martínez-Bernal, Susan Morey, and Rafael H. Villarreal, Associated primes of powers of edge ideals, Collect. Math. 63 (2012), no. 3, 361–374. MR 2957976, DOI 10.1007/s13348-011-0045-9
Susan Morey, Depths of powers of the edge ideal of a tree, Comm. Algebra 38 (2010), no. 11, 4042–4055. MR 2764849, DOI 10.1080/00927870903286900
Susan Morey and Rafael H. Villarreal, Edge ideals: algebraic and combinatorial properties, Progress in commutative algebra 1, de Gruyter, Berlin, 2012, pp. 85–126. MR 2932582
Giancarlo Rinaldo, Naoki Terai, and Ken-ichi Yoshida, Cohen-Macaulayness for symbolic power ideals of edge ideals, J. Algebra 347 (2011), 1–22. MR 2846393, DOI 10.1016/j.jalgebra.2011.09.007
H. E. Robbins, Questions, Discussions, and Notes: A Theorem on Graphs, with an Application to a Problem of Traffic Control, Amer. Math. Monthly 46 (1939), no. 5, 281–283. MR 1524589, DOI 10.2307/2303897
Rodney Y. Sharp, Convergence of sequences of sets of associated primes, Proc. Amer. Math. Soc. 131 (2003), no. 10, 3009–3017. MR 1993206, DOI 10.1090/S0002-9939-03-07038-2
Aron Simis, Wolmer V. Vasconcelos, and Rafael H. Villarreal, On the ideal theory of graphs, J. Algebra 167 (1994), no. 2, 389–416. MR 1283294, DOI 10.1006/jabr.1994.1192
Patrick Solé and Thomas Zaslavsky, The covering radius of the cycle code of a graph, Discrete Appl. Math. 45 (1993), no. 1, 63–70. MR 1230281, DOI 10.1016/0166-218X(93)90140-J
Yukihide Takayama, Combinatorial characterizations of generalized Cohen-Macaulay monomial ideals, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 48(96) (2005), no. 3, 327–344. MR 2165349
Naoki Terai and Ngo Viet Trung, On the associated primes and the depth of the second power of squarefree monomial ideals, J. Pure Appl. Algebra 218 (2014), no. 6, 1117–1129. MR 3153618, DOI 10.1016/j.jpaa.2013.11.008
Tran Nam Trung, Stability of depths of powers of edge ideals, J. Algebra 452 (2016), 157–187. MR 3461061, DOI 10.1016/j.jalgebra.2016.01.009
Douglas B. West, Introduction to graph theory, Prentice Hall, Inc., Upper Saddle River, NJ, 1996. MR 1367739