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Regularity of symbolic powers and arboricity of matroids

Nguyên Công Minh and Trân Nam Trung
From the journal Forum Mathematicum

Abstract

Let Δ be a matroid complex. In this paper, we explicitly compute the regularity of all the symbolic powers of its Stanley–Reisner ideal in terms of combinatorial data of Δ. In order to do that, we provide a sharp bound between the arboricity of Δ and the circumference of its dual Δ*.

MSC 2010: 13D45; 05E40; 05E45

Communicated by Jan Bruinier


Funding statement: This paper was done while the first author was visiting the Vietnam Institute for Advanced Study in Mathematics (VIASM). He would like to thank the VIASM for the hospitality and financial support, and he also thanks the Vietnam National Foundation for Science and Technology Development (NAFOSTED) for its support under grant number 101.01-2016.21.

Acknowledgements

We also thank the two anonymous referees of this paper for their very useful corrections and suggestions.

References

[1] N. Alon, C. McDiarmid and B. Reed, Star arboricity, Combinatorica 12 (1992), no. 4, 375–380. 10.1007/BF01305230Search in Google Scholar

[2] S. Beyarslan, H. T. Hà and T. N. Trung, Regularity of powers of forests and cycles, J. Algebraic Combin. 42 (2015), no. 4, 1077–1095. 10.1007/s10801-015-0617-ySearch in Google Scholar

[3] M. Chardin, Powers of ideals and the cohomology of stalks and fibers of morphisms, Algebra Number Theory 7 (2013), no. 1, 1–18. 10.2140/ant.2013.7.1Search in Google Scholar

[4] S. D. Cutkosky, Irrational asymptotic behaviour of Castelnuovo–Mumford regularity, J. Reine Angew. Math. 522 (2000), 93–103. 10.1515/crll.2000.043Search in Google Scholar

[5] S. D. Cutkosky, J. Herzog and N. V. Trung, Asymptotic behaviour of the Castelnuovo–Mumford regularity, Compos. Math. 118 (1999), no. 3, 243–261. 10.1023/A:1001559912258Search in Google Scholar

[6] H. Dao, A. De Stefani, E. Grifo, C. Huneke and L. Núñez Betancourt, Symbolic powers of ideals, Singularities and Foliations. Geometry, Topology and Applications, Springer Proc. Math. Stat. 222, Springer, Cham (2018), 387–432. 10.1007/978-3-319-73639-6_13Search in Google Scholar

[7] A. M. Dean, J. P. Hutchinson and E. R. Scheinerman, On the thickness and arboricity of a graph, J. Combin. Theory Ser. B 52 (1991), no. 1, 147–151. 10.1016/0095-8956(91)90100-XSearch in Google Scholar

[8] J. Edmonds, Minimum partition of a matroid into independent subsets, J. Res. Nat. Bur. Standards Sect. B 69B (1965), 67–72. 10.6028/jres.069B.004Search in Google Scholar

[9] D. Eisenbud, The Geometry of Syzygies, Grad. Texts in Math. 229, Springer, New York, 2005. Search in Google Scholar

[10] D. Eisenbud and J. Harris, Powers of ideals and fibers of morphisms, Math. Res. Lett. 17 (2010), no. 2, 267–273. 10.4310/MRL.2010.v17.n2.a6Search in Google Scholar

[11] D. Eisenbud and B. Ulrich, Notes on regularity stabilization, Proc. Amer. Math. Soc. 140 (2012), no. 4, 1221–1232. 10.1090/S0002-9939-2011-11270-XSearch in Google Scholar

[12] J. Herzog, T. Hibi and N. V. Trung, Symbolic powers of monomial ideals and vertex cover algebras, Adv. Math. 210 (2007), no. 1, 304–322. 10.1016/j.aim.2006.06.007Search in Google Scholar

[13] J. Herzog, L. T. Hoa and N. V. Trung, Asymptotic linear bounds for the Castelnuovo–Mumford regularity, Trans. Amer. Math. Soc. 354 (2002), no. 5, 1793–1809. 10.1090/S0002-9947-02-02932-XSearch in Google Scholar

[14] T. Hibi, Buchsbaum complexes with linear resolutions, J. Algebra 179 (1996), no. 1, 127–136. 10.1006/jabr.1996.0006Search in Google Scholar

[15] L. T. Hoa and T. N. Trung, Partial Castelnuovo–Mumford regularities of sums and intersections of powers of monomial ideals, Math. Proc. Cambridge Philos. Soc. 149 (2010), no. 2, 229–246. 10.1017/S0305004110000071Search in Google Scholar

[16] L. T. Hoa and T. N. Trung, Castelnuovo–Mumford regularity of symbolic powers of two-dimensional square-free monomial ideals, J. Commut. Algebra 8 (2016), no. 1, 77–88. 10.1216/JCA-2016-8-1-77Search in Google Scholar

[17] V. Kodiyalam, Asymptotic behaviour of Castelnuovo–Mumford regularity, Proc. Amer. Math. Soc. 128 (2000), no. 2, 407–411. 10.1090/S0002-9939-99-05020-0Search in Google Scholar

[18] N. C. Minh, N. Terai and P. T. Thuy, Level property of ordinary and symbolic powers of Stanley–Reisner ideals, preprint (2018). 10.1016/j.jalgebra.2019.05.044Search in Google Scholar

[19] N. C. Minh and N. V. Trung, Cohen–Macaulayness of monomial ideals and symbolic powers of Stanley–Reisner ideals, Adv. Math. 226 (2011), no. 2, 1285–1306. 10.1016/j.aim.2010.08.005Search in Google Scholar

[20] C. S. J. A. Nash-Williams, Decomposition of finite graphs into forests, J. Lond. Math. Soc. 39 (1964), 1–12. 10.1112/jlms/s1-39.1.12Search in Google Scholar

[21] J. G. Oxley, Matroid Theory, Clarendon Press, New York, 1992. Search in Google Scholar

[22] P. D. Seymour, A note on list arboricity, J. Combin. Theory Ser. B 72 (1998), no. 1, 150–151. 10.1006/jctb.1997.1784Search in Google Scholar

[23] R. P. Stanley, Combinatorics and Commutative Algebra, 2nd ed., Progr. Math. 41, Birkhäuser, Boston, 1996. Search in Google Scholar

[24] Y. Takayama, Combinatorial characterizations of generalized Cohen–Macaulay monomial ideals, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 48(96) (2005), no. 3, 327–344. Search in Google Scholar

[25] N. Terai and N. V. Trung, Cohen–Macaulayness of large powers of Stanley–Reisner ideals, Adv. Math. 229 (2012), no. 2, 711–730. 10.1016/j.aim.2011.10.004Search in Google Scholar

[26] N. V. Trung and H.-J. Wang, On the asymptotic linearity of Castelnuovo–Mumford regularity, J. Pure Appl. Algebra 201 (2005), no. 1–3, 42–48. 10.1016/j.jpaa.2004.12.043Search in Google Scholar

[27] M. Varbaro, Symbolic powers and matroids, Proc. Amer. Math. Soc. 139 (2011), no. 7, 2357–2366. 10.1090/S0002-9939-2010-10685-8Search in Google Scholar

[28] H. Whitney, On the abstract properties of linear dependence, Amer. J. Math. 57 (1935), no. 3, 509–533. 10.2307/2371182Search in Google Scholar

Received: 2017-11-21
Revised: 2018-09-28
Published Online: 2018-10-30
Published in Print: 2019-03-01

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