Acta Mathematica Vietnamica

, Volume 44, Issue 1, pp 243–268 | Cite as

Depth and Regularity of Monomial Ideals via Polarization and Combinatorial Optimization

  • José Martínez-Bernal
  • Susan Morey
  • Rafael H. VillarrealEmail author
  • Carlos E. Vivares


In this paper, we use polarization to study the behavior of the depth and regularity of a monomial ideal I, locally at a variable xi, when we lower the degree of all the highest powers of the variable xi occurring in the minimal generating set of I, and examine the depth and regularity of powers of edge ideals of clutters using combinatorial optimization techniques. If I is the edge ideal of an unmixed clutter with the max-flow min-cut property, we show that the powers of I have non-increasing depth and non-decreasing regularity. In particular, edge ideals of unmixed bipartite graphs have non-decreasing regularity. We are able to show that the symbolic powers of the ideal of covers of the clique clutter of a strongly perfect graph have non-increasing depth. A similar result holds for the ideal of covers of a uniform ideal clutter.


Depth Regularity Max-flow min-cut Clutter Edge ideal Monomial ideal Polarization 

Mathematics Subject Classification (2010)

Primary 13F20 Secondary 05C22 05E40 13H10 



We thank the referee for a careful reading of the paper and for the improvements suggested.

Funding Information

The first and third authors were partially supported by SNI. The fourth author was supported by a scholarship from CONACYT.


  1. 1.
    Brodmann, M.: The asymptotic nature of the analytic spread. Math. Proc. Cambridge Philos. Soc. 86, 35–39 (1979)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Burch, L.: Codimension and analytic spread. Proc. Cambridge Philos. Soc. 72, 369–373 (1972)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Caviglia, G., Hà, H.T., Herzog, J., Kummini, M., Terai, N., Trung, N.V.: Depth and regularity modulo a principal ideal. J. Algebraic Comb., to appearGoogle Scholar
  4. 4.
    Chen, J., Morey, S., Sung, A.: The stable set of associated primes of the ideal of a graph. Rocky Mountain J. Math. 32(1), 71–89 (2002)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Constantinescu, A., Pournaki, M.R., Seyed Fakhari, S.A., Terai, N., Yassemi, S.: Cohen-Macaulayness and limit behavior of depth for powers of cover ideals. Commun. Algebra 43(1), 143–157 (2015)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Cornuéjols, G.: Combinatorial Optimization. Packing and Covering. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 74. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2001)Google Scholar
  7. 7.
    Crupi, M., Rinaldo, G., Terai, N., Yoshida, K.: Effective Cowsik–Nori theorem for edge ideals. Commun. Algebra 38(9), 3347–3357 (2010)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Dao, H., De Stefani, A., Grifo, E., Huneke, C., Núñez-Betancourt, L.: Symbolic Powers of Ideals. Singularities and Foliations, Geometry, Topology and Applications, 387–432. Springer Proc. Math. Stat., vol. 222. Springer, Cham (2018)Google Scholar
  9. 9.
    Dao, H., Huneke, C., Schweig, J.: Bounds on the regularity and projective dimension of ideals associated to graphs. J. Algebraic Comb. 38(1), 37–55 (2013)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Dupont, L.A., Villarreal, R.H.: Algebraic and combinatorial properties of ideals and algebras of uniform clutters of TDI systems. J. Comb. Optim. 21(3), 269–292 (2011)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Eisenbud, D.: Commutative Algebra with a View Toward Algebraic Geometry. Graduate Texts in Mathematics, vol. 150. Springer, Berlin (1995)Google Scholar
  12. 12.
    Eisenbud, D., Huneke, C.: Cohen–Macaulay Rees algebras and their specialization. J. Algebra 81, 202–224 (1983)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Escobar, C., Villarreal, R.H., Yoshino, Y.: Torsion Freeness and Normality of Blowup Rings of Monomial Ideals. Commutative Algebra. Lect. Notes Pure Appl. Math., vol. 244, pp 69–84. Chapman & Hall/CRC, Boca Raton (2006)zbMATHGoogle Scholar
  14. 14.
    Faridi, S.: Monomial Ideals via Square-Free Monomial Ideals. Lecture Notes in Pure and Applied Math, vol. 244, pp 85–114. Taylor & Francis, Philadelphia (2005)Google Scholar
  15. 15.
    Francisco, C., Hà, H.T., Mermin, J.: Powers of Square-Free Monomial Ideals and Combinatorics. Commutative Algebra, pp 373–392. Springer, New York (2013)zbMATHGoogle Scholar
  16. 16.
    Fröberg, R.: A study of graded extremal rings and of monomial rings. Math. Scand. 51, 22–34 (1982)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Gimenez, P., Martínez-Bernal, J., Simis, A., Villarreal, R.H., Vivares, C.R.: Symbolic powers of monomial ideals and Cohen–Macaulay vertex-weighted digraphs. Special volume dedicated to Antonio Campillo, Springer, to appearGoogle Scholar
  18. 18.
    Gitler, I., Reyes, E., Villarreal, R.H.: Blowup algebras of square-free monomial ideals and some links to combinatorial optimization problems. Rocky Mountain J. Math. 39(1), 71–102 (2009)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Gitler, I., Valencia, C.E.: On bounds for some graph invariants. Bol. Soc. Mat. Mexicana 3, 16(2), 73–94 (2010)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Gitler, I., Valencia, C., Villarreal, R.H.: A note on Rees algebras and the MFMC property. Beiträge Algebra Geom. 48(1), 141–150 (2007)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Gitler, I., Villarreal, R.H.: Graphs, Rings and Polyhedra. Aportaciones Mat. Textos, vol. 35. Soc. Mat. Mexicana, México (2011)Google Scholar
  22. 22.
    Grayson, D., Stillman, M.: Macaulay2. Available via anonymous ftp from (1996)
  23. 23.
    Hà, H.T., Lin, K.-N., Morey, S., Reyes, E., Villarreal, R.H.: Edge ideals of oriented graphs. Internat. J. Algebra Comput., to appear (2018)Google Scholar
  24. 24.
    Hà, H.T., Morey, S.: Embedded associated primes of powers of square-free monomial ideals. J. Pure Appl. Algebra 214(4), 301–308 (2010)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Hà, H.T., Trung, N.V., Trung, T.N.: Depth and regularity of powers of sums of ideals. Math. Z. 282(3–4), 819–838 (2016)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Hang, N.T.: Stability of depth functions of cover ideals of balanced hypergraphs. Preprint, arXiv:1711.09178 (2017)
  27. 27.
    Hang, N.T., Trung, T.N.: The behavior of depth functions of cover ideals of unimodular hypergraphs. Ark. Mat. 55(1), 89–104 (2017)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Harary, F.: Graph Theory. Addison-Wesley, Reading (1972)zbMATHGoogle Scholar
  29. 29.
    Herzog, J., Hibi, T.: Distributive lattices, bipartite graphs and Alexander duality. J. Algebraic Combin. 22(3), 289–302 (2005)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Herzog, J., Hibi, T.: The depth of powers of an ideal. J. Algebra 291, 534–550 (2005)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Herzog, J., Hibi, T.: Monomial Ideals. Graduate Texts in Mathematics, vol. 260. Springer, Berlin (2011)Google Scholar
  32. 32.
    Herzog, J., Takayama, Y., Terai, N.: On the radical of a monomial ideal. Arch. Math. 85, 397–408 (2005)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Hoa, L.T., Kimura, K., Terai, N., Trung, T.N.: Stability of depths of symbolic powers of Stanley-Reisner ideals. J. Algebra 473, 307–323 (2017)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Hoa, L.T., Tam, N.D.: On some invariants of a mixed product of ideals. Arch. Math. (Basel) 94(4), 327–337 (2010)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Hoang, D.T., Minh, N.C., Trung, T.N.: Combinatorial characterizations of the Cohen-Macaulayness of the second power of edge ideals. J. Comb. Theory Ser. A 120(5), 1073–1086 (2013)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Hoang, D.T., Trung, T.N.: A characterization of triangle-free Gorenstein graphs and Cohen-Macaulayness of second powers of edge ideals. J. Algebraic Comb. 43(2), 325–338 (2016)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Jayanthan, A.V., Narayanan, N., Selvaraja, S.: Regularity of powers of bipartite graphs. J. Algebraic Comb. 47(1), 17–38 (2018)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Kaiser, T., Stehlík, M., Škrekovski, R.: Replication in critical graphs and the persistence of monomial ideals. J. Comb. Theory Ser. A 123(1), 239–251 (2014)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Kimura, K., Terai, N., Yassemi, S.: The projective dimension of the edge ideal of a very well-covered graph. Nagoya Math. J. 230, 160–179 (2018)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Martínez-Bernal, J., Morey, S., Villarreal, R.H.: Associated primes of powers of edge ideals. Collect. Math. 63(3), 361–374 (2012)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Martínez-Bernal, J., Pitones, Y., Villarreal, R.H.: Minimum distance functions of graded ideals and Reed-Muller-type codes. J. Pure Appl. Algebra 221, 251–275 (2017)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Matsumura, H.: Commutative Ring Theory. Cambridge Studies in Advanced Mathematics, vol. 8. Cambridge University Press, Cambridge (1986)Google Scholar
  43. 43.
    Morey, S., Villarreal, R.H.: Edge ideals: algebraic and combinatorial properties. In: Francisco, C., Klingler, L.C., Sather-Wagstaff, S., Vassilev J.C. (eds.) Progress in Commutative Algebra, Combinatorics and Homology, vol. 1, pp 85–126. De Gruyter, Berlin (2012)Google Scholar
  44. 44.
    Neves, J., Vaz Pinto, M., Villarreal, R.H.: Regularity and algebraic properties of certain lattice ideals. Bull. Braz. Math. Soc. (N.S.) 45, 777–806 (2014)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Peeva, I.: Graded Syzygies. Algebra and Applications, vol. 14. Springer, Berlin (2011)zbMATHGoogle Scholar
  46. 46.
    Ravi, M.S.: Regularity of ideals and their radicals. Manuscripta Math. 68(1), 77–87 (1990)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Ravindra, G.: Some classes of strongly perfect graphs. Discrete Math. 206(1–3), 197–203 (1999)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Rinaldo, G., Terai, N., Yoshida, K.: On the second powers of Stanley-Reisner ideals. J. Commut. Algebra 3(3), 405–430 (2011)MathSciNetzbMATHGoogle Scholar
  49. 49.
    Rinaldo, G., Terai, N., Yoshida, Y.: Cohen-Macaulayness for symbolic power ideals of edge ideals. J. Algebra 347, 1–22 (2011)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Schrijver, A.: Theory of Linear and Integer Programming. Wiley, New York (1986)zbMATHGoogle Scholar
  51. 51.
    Schrijver, A.: Combinatorial Optimization Algorithms and Combinatorics, vol. 24. Springer, Berlin (2003)Google Scholar
  52. 52.
    Seyed Fakhari, S.A.: Symbolic powers of cover ideal of very well-covered and bipartite graphs. Preprint, arXiv:1604.00654v1 (2016)
  53. 53.
    Seyed Fakhari, S.A.: Depth and Stanley depth of symbolic powers of cover ideals of graphs. J. Algebra 492, 402–413 (2017)MathSciNetzbMATHGoogle Scholar
  54. 54.
    Seyed Fakhari, S.A.: Symbolic powers of cover ideal of very well-covered and bipartite graphs. Proc. Am. Math. Soc. 146(1), 97–110 (2018)MathSciNetzbMATHGoogle Scholar
  55. 55.
    Simis, A., Vasconcelos, W.V., Villarreal, R.H.: On the ideal theory of graphs. J. Algebra 167, 389–416 (1994)MathSciNetzbMATHGoogle Scholar
  56. 56.
    Smith, D.E.: On the Cohen–Macaulay property in commutative algebra and simplicial topology. Pacific J. Math. 141, 165–196 (1990)MathSciNetzbMATHGoogle Scholar
  57. 57.
    Sørensen, A.: Projective Reed-Muller codes. IEEE Trans. Inform. Theory 37(6), 1567–1576 (1991)MathSciNetzbMATHGoogle Scholar
  58. 58.
    Terai, N.: Alexander duality theorem and Stanley–Reisner rings. Sūrikaisekikenkyūsho Kōkyūroku 1078, 174–184 (1999)MathSciNetzbMATHGoogle Scholar
  59. 59.
    Terai, N., Trung, N.V.: Cohen-Macaulayness of large powers of Stanley-Reisner ideals. Adv. Math. 229(2), 711–730 (2012)MathSciNetzbMATHGoogle Scholar
  60. 60.
    Trung, N.V., Tuan, T.M.: Equality of ordinary and symbolic powers of Stanley-Reisner ideals. J. Algebra 328, 77–93 (2011)MathSciNetzbMATHGoogle Scholar
  61. 61.
    Trung, T.N.: Stability of depths of powers of edge ideals. J. Algebra 452, 157–187 (2016)MathSciNetzbMATHGoogle Scholar
  62. 62.
    Van Tuyl, A.: A beginner’s guide to edge and cover ideals. In: Monomial Ideals, Computations and Applications. Lecture Notes in Mathematics, vol. 2083, pp 63–94. Springer (2013)Google Scholar
  63. 63.
    Vasconcelos, W.V.: Arithmetic of Blowup Algebras. London Math. Soc. Lecture Note Series, vol. 195. Cambridge University Press, Cambridge (1994)Google Scholar
  64. 64.
    Vasconcelos, W.V.: Computational Methods in Commutative Algebra and Algebraic Geometry. Springer, Berlin (1998)zbMATHGoogle Scholar
  65. 65.
    Villarreal, R.H.: Cohen–Macaulay graphs. Manuscripta Math. 66, 277–293 (1990)MathSciNetzbMATHGoogle Scholar
  66. 66.
    Villarreal, R.H.: Monomial Algebras. Monographs and Research Notes in Mathematics, 2nd edn. Chapman and Hall/CRC, London (2015)Google Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Departamento de MatemáticasCentro de Investigación y de Estudios Avanzados del IPNMexico CityMexico
  2. 2.Department of MathematicsTexas State UniversitySan MarcosUSA

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