Betti numbers associated to the facet ideal of a matroid

  • Trygve Johnsen
  • Jan Roksvold
  • Hugues Verdure


To a matroid M with n edges, we associate the so-called facet ideal \(\mathcal{F}(M) \subset \Bbbk [x_1 , \ldots ,x_n ]\), generated by monomials corresponding to bases of M. We show that when M is a graph, the Betti numbers related to an ℕ0-graded minimal free resolution of \(\mathcal{F}(M)\) are determined by the Betti numbers related to the blocks of M. Similarly, we show that the higher weight hierarchy of M is determined by the weight hierarchies of the blocks, as well. Drawing on these results, we show that when M is the cycle matroid of a cactus graph, the Betti numbers determine the higher weight hierarchy — and vice versa. Finally, we demonstrate by way of counterexamples that this fails to hold for outerplanar graphs in general.


matroids facet ideals Betti numbers higher weights blocks cactus graphs 

Mathematical subject classification

05B35 05E40 13D02 05C25 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    J.A. Eagon and V. Reiner. Resolutions of Stanley-Reisner rings and Alexander duality. Journal of Pure and Applied Algebra, 130 (1998), 265–275.CrossRefMathSciNetzbMATHGoogle Scholar
  2. [2]
    S. Faridi. The facet ideal of a simplicial complex. ManuscriptaMath., 109(2) (2002), 159–174.CrossRefMathSciNetzbMATHGoogle Scholar
  3. [3]
    T. Johnsen and H. Verdure. Hamming weights and Betti numbers of Stanley-Reisner rings associated to matroids. Applicable Algebra in Engineering, Communication and Computing, 24(1) (2013), 73–93.CrossRefMathSciNetzbMATHGoogle Scholar
  4. [4]
    E. Miller and B. Sturmfels. Combinatorial Commutative Algebra. Graduate Texts in Mathematics, 227 (2005), Springer-Verlag.Google Scholar
  5. [5]
    D.G. Northcott. Homological Algebra. Cambridge University Press (1966).Google Scholar
  6. [6]
    J.G. Oxley. Matroid Theory. Oxford University Press (1992).zbMATHGoogle Scholar
  7. [7]
    V.K. Wei. Generalized Hamming Weights for Linear Codes. IEEE Transactions on Information Theory, 37 (1991), 1412–1418.CrossRefzbMATHGoogle Scholar

Copyright information

© Sociedade Brasileira de Matemática 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TromsøTromsøNorway

Personalised recommendations