Journal of Commutative Algebra

Monomial resolutions supported by simplicial trees

Sara Faridi

Full-text: Open access


We explore resolutions of monomial ideals supported by simplicial trees. We argue that, since simplicial trees are acyclic, the criterion of Bayer, Peeva and Sturmfels for checking whether a simplicial complex supports a free resolution of a monomial ideal reduces to checking that certain induced subcomplexes are connected. We then use results of Peeva and Velasco to show that every simplicial tree appears as the Scarf complex of a monomial ideal and hence supports a minimal resolution. We also provide a way to construct smaller Scarf ideals than those constructed by Peeva and Velasco.

Article information

J. Commut. Algebra, Volume 6, Number 3 (2014), 347-361.

First available in Project Euclid: 17 November 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Faridi, Sara. Monomial resolutions supported by simplicial trees. J. Commut. Algebra 6 (2014), no. 3, 347--361. doi:10.1216/JCA-2014-6-3-347.

Export citation


  • D. Bayer, I. Peeva and B. Sturmfels, Monomial resolutions, Math. Res. Lett. 5 (1998), 31–46.
  • A. Björner, Topological methods, Handbook of combinatorics, Volumes 1 and 2, Elsevier, Amsterdam, 1995.
  • M. Caboara and S. Faridi, Odd-cycle-free complexes and the Koenig property, Rocky Mountain J. Math. 41 (2011), 1059–1079.
  • M. Caboara, S. Faridi and P. Selinger, Simplicial cycles and the computation of simplicial trees, J. Sym. Comp. 42 (2007), 74–88.
  • S. Faridi, The facet ideal of a simplicial complex, Manuscr. Math. 109 (2002), 159–174.
  • ––––, Cohen-Macaulay properties of square-free monomial ideals, J. Comb. Theor. 109 (2005), 299–329.
  • ––––, Simplicial trees are sequentially Cohen-Macaulay, J. Pure Appl. Alg. 190, (2004), 121–136.
  • V. Gasharov, I. Peeva and V. Welker, The lcm-lattice in monomial resolutions, Math. Res. Lett. 6 (1999), 521–532.
  • I. Peeva and M. Velasco, Frames and degenerations of monomial resolutions, Trans. Amer. Math. Soc. 363 (2011), 2029–2046.
  • J. Phan, Minimal monomial ideals and linear resolutions. (2005).
  • D. Taylor, Ideals generated by monomials in an $R$-sequence, Ph.D. thesis, University of Chicago, 1966.
  • M. Velasco, Minimal free resolutions that are not supported by a CW-complex, J. Alg. 319 (2008), 102–114.