On Linear Quotients of Squarefree Monomial Ideals

  • Erfan Manouchehri
  • Ali Soleyman JahanEmail author


Let \(I\subset K[x_1,\ldots ,x_n]\) be a squarefree monomial ideal generated in one degree. Let \(G_I\) be the graph whose nodes are the generators of I, and two vertices \(u_i\) and \(u_{ j}\) are adjacent if there exist variables xy such that \(xu_i = yu_j\). We show that if I is generated in degree \(n-2\), then the following are equivalent:
  1. (i)

    \(G_I\) is a connected graph;

  2. (ii)

    I has a \((n-2)\)-linear resolution;

  3. (iii)

    I has linear quotients;

  4. (iv)

    I is a variable-decomposable ideal.

We also prove that if I has linear relations and \(\overline{G_I}\) is chordal, then I has linear quotients.


Linear syzygies Linear resolution Linear quotients Variable-decomposable 



  1. 1.
    Ajdani, S.M., Jahan, A.S.: Vertex decomposability of \(2\)-CM and Gorenstein simplicial complexes of codimension \(3\). Bull. Malays. Math. Sci. Soc. 39, 609–617 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Chartrand, G., Zhang, P.: A First Course in Graph Theory. Courier Corporation (2013)Google Scholar
  3. 3.
    Eagon, J.A., Reiner, V.: Resolutions of Stanley–Reisner rings and Alexander duality. J. Pure Appl. Algebra 130, 265–275 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Emtander, E.: A class of hypergraphs that generalizes chordal graphs. Math. Scand. 106(1), 50–66 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Fröberg, R.: On Stanley–Reisner rings. Top. Algebra Banach Cent. Publ. 26, 57–70 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Herzog, J., Hibi, T.: Monomial Ideals, Graduate Texts in Mathematics, vol. 260. Springer, London (2011)CrossRefzbMATHGoogle Scholar
  7. 7.
    Herzog, J., Hibi, T., Zheng, X.: Monomial ideals whose powers have a linear resolution. Math. Scand. 95, 23–32 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Manouchehri, E., Jahan, A.S.: On Linear Resolution and Linear Quotients of Monomial Ideals. arXiv:1809.00133
  9. 9.
    Moriyama, S., Takeuchi, F.: Inccremental construction properties in dimension two: shellability, extendable shellability and vertex decomposability. Discrete Math. 263, 295–296 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Rahim Rahmati, A., Yassemi, S.: \(k\)-Decomposable monomial ideals. Algebra Colloq. 22, 745–756 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Tuyl, A., Villarreal, R.: Shellable graphs and sequentially Cohen–Macaulay bipartite graphs. J. Combin. Theory Ser. A. 115, 799–814 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Woodroofe, R.: Chordal and sequentially Cohen–Macaulay clutters. Electron. J. Combin. 18, P208 (2011)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Yazdan Pour, A.A.: Resolutions and Castelnuovo–Mumford Regularity, Differential Geometry [math.DG]. Université de Grenoble (2012)Google Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of KurdistanSanandajIran

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