Rocky Mountain Journal of Mathematics

Expansions of monomial ideals and multigraded modules

Shamila Bayati and Jürgen Herzog

Full-text: Open access


We introduce an exact functor defined on multigraded modules which we call the expansion functor and study its homological properties. The expansion functor applied to a monomial ideal amounts to substitute the variables by monomial prime ideals and to apply this substitution to the generators of the ideal. This operation naturally occurs in various combinatorial contexts.

Article information

Rocky Mountain J. Math., Volume 44, Number 6 (2014), 1781-1804.

First available in Project Euclid: 2 February 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 13C13: Other special types 13D02: Syzygies, resolutions, complexes

Expansion functor free resolution graded Betti numbers monomial ideals


Bayati, Shamila; Herzog, Jürgen. Expansions of monomial ideals and multigraded modules. Rocky Mountain J. Math. 44 (2014), no. 6, 1781--1804. doi:10.1216/RMJ-2014-44-6-1781.

Export citation


  • M. Brodmann, Asymptotic stability of ${\Ass}(M/I^{n}M)$, Proc. Amer. Math. Soc. 74 (1979), 16–18.
  • W. Bruns and U. Vetter, Determinantal rings, Lect. Notes Math. 1327, Springer, New York, 1988.
  • A. Conca and J. Herzog, Castelnuovo-Mumford regularity of products of ideals, Collect. Math. 54 (2003), 137–152.
  • R. Diestel, Graph theory, Second edition, Grad. Texts Math. 173, Springer-Verlag, New York, 2000.
  • S.P. Dutta, Syzygies and homological conjectures, Commutative algebra, Math. Sci. Res. Inst. Publ. 15, Springer, New York, 1989.
  • C.A. Francisco, H. Tài Hà and A. Van Tuyl, Colorings of hypergraphs, perfect graphs, and associated primes of powers of monomial ideals, J. Algebra 331 (2011), 224–242.
  • J. Herzog, T. Hibi, N.V. Trung and X. Zheng, Standard graded vertex cover algebras, cycles and leaves, Trans. Amer. Math. Soc. 360 (2008), 6231–6249.
  • J. Herzog and M. Kühl, On the Betti numbers of finite pure and linear resolutions, Comm. Algebra 12 (1984), no. 13-14, 1627–1646.
  • J. Herzog and Y. Takayama, Resolutions by mapping cones, The Roos Festschrift 4, Homology, Homotopy and Applications, (2002), 277–294.
  • J. Martínez-Bernal, S. Morey and R.H. Villarreal, Associated primes of powers of edge ideals, Collect. Math. 63 (2012), 361–374.
  • J.J. Rotman, An introduction to homological algebra, Second edition, Universitext., Springer, New York, 2009.