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Betti numbers associated to the facet ideal of a matroid

Abstract

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.

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Correspondence to Jan Roksvold.

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Dedicated to Steven L. Kleiman and Aron Simis on occasion of their 70th birthdays

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Johnsen, T., Roksvold, J. & Verdure, H. Betti numbers associated to the facet ideal of a matroid. Bull Braz Math Soc, New Series 45, 727–744 (2014). https://doi.org/10.1007/s00574-014-0071-9

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Keywords

  • matroids
  • facet ideals
  • Betti numbers
  • higher weights
  • blocks
  • cactus graphs

Mathematical subject classification

  • 05B35
  • 05E40
  • 13D02
  • 05C25