Depth and Regularity of Monomial Ideals via Polarization and Combinatorial Optimization

Abstract

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.

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References

  1. 1.

    Brodmann, M.: The asymptotic nature of the analytic spread. Math. Proc. Cambridge Philos. Soc. 86, 35–39 (1979)

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    Burch, L.: Codimension and analytic spread. Proc. Cambridge Philos. Soc. 72, 369–373 (1972)

    MathSciNet  MATH  Article  Google 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 appear

  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)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google 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)

    Google 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)

    Google Scholar 

  16. 16.

    Fröberg, R.: A study of graded extremal rings and of monomial rings. Math. Scand. 51, 22–34 (1982)

    MathSciNet  MATH  Article  Google 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 appear

  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)

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Gitler, I., Valencia, C.E.: On bounds for some graph invariants. Bol. Soc. Mat. Mexicana 3, 16(2), 73–94 (2010)

    MathSciNet  MATH  Google 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)

    MathSciNet  MATH  Google 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 math.uiuc.edu (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)

  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)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google Scholar 

  28. 28.

    Harary, F.: Graph Theory. Addison-Wesley, Reading (1972)

    Google Scholar 

  29. 29.

    Herzog, J., Hibi, T.: Distributive lattices, bipartite graphs and Alexander duality. J. Algebraic Combin. 22(3), 289–302 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  30. 30.

    Herzog, J., Hibi, T.: The depth of powers of an ideal. J. Algebra 291, 534–550 (2005)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google Scholar 

  37. 37.

    Jayanthan, A.V., Narayanan, N., Selvaraja, S.: Regularity of powers of bipartite graphs. J. Algebraic Comb. 47(1), 17–38 (2018)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google 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)

  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)

    MathSciNet  MATH  Article  Google Scholar 

  45. 45.

    Peeva, I.: Graded Syzygies. Algebra and Applications, vol. 14. Springer, Berlin (2011)

    Google Scholar 

  46. 46.

    Ravi, M.S.: Regularity of ideals and their radicals. Manuscripta Math. 68(1), 77–87 (1990)

    MathSciNet  MATH  Article  Google Scholar 

  47. 47.

    Ravindra, G.: Some classes of strongly perfect graphs. Discrete Math. 206(1–3), 197–203 (1999)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google Scholar 

  49. 49.

    Rinaldo, G., Terai, N., Yoshida, Y.: Cohen-Macaulayness for symbolic power ideals of edge ideals. J. Algebra 347, 1–22 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  50. 50.

    Schrijver, A.: Theory of Linear and Integer Programming. Wiley, New York (1986)

    Google 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)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google Scholar 

  55. 55.

    Simis, A., Vasconcelos, W.V., Villarreal, R.H.: On the ideal theory of graphs. J. Algebra 167, 389–416 (1994)

    MathSciNet  MATH  Article  Google Scholar 

  56. 56.

    Smith, D.E.: On the Cohen–Macaulay property in commutative algebra and simplicial topology. Pacific J. Math. 141, 165–196 (1990)

    MathSciNet  MATH  Article  Google Scholar 

  57. 57.

    Sørensen, A.: Projective Reed-Muller codes. IEEE Trans. Inform. Theory 37(6), 1567–1576 (1991)

    MathSciNet  MATH  Article  Google Scholar 

  58. 58.

    Terai, N.: Alexander duality theorem and Stanley–Reisner rings. Sūrikaisekikenkyūsho Kōkyūroku 1078, 174–184 (1999)

    MathSciNet  MATH  Google Scholar 

  59. 59.

    Terai, N., Trung, N.V.: Cohen-Macaulayness of large powers of Stanley-Reisner ideals. Adv. Math. 229(2), 711–730 (2012)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google Scholar 

  61. 61.

    Trung, T.N.: Stability of depths of powers of edge ideals. J. Algebra 452, 157–187 (2016)

    MathSciNet  MATH  Article  Google 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)

  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)

    Google Scholar 

  65. 65.

    Villarreal, R.H.: Cohen–Macaulay graphs. Manuscripta Math. 66, 277–293 (1990)

    MathSciNet  MATH  Article  Google Scholar 

  66. 66.

    Villarreal, R.H.: Monomial Algebras. Monographs and Research Notes in Mathematics, 2nd edn. Chapman and Hall/CRC, London (2015)

    Google Scholar 

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Acknowledgements

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

Funding

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

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Correspondence to Rafael H. Villarreal.

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Martínez-Bernal, J., Morey, S., Villarreal, R.H. et al. Depth and Regularity of Monomial Ideals via Polarization and Combinatorial Optimization. Acta Math Vietnam 44, 243–268 (2019). https://doi.org/10.1007/s40306-018-00308-z

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Keywords

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

Mathematics Subject Classification (2010)

  • Primary 13F20
  • Secondary 05C22
  • 05E40
  • 13H10