Journal of Algebraic Combinatorics

, Volume 39, Issue 3, pp 711–718 | Cite as

Betti tables of p-Borel-fixed ideals



In this note we provide a counterexample to a conjecture of Pardue (Thesis (Ph.D.), Brandeis University, 1994), which asserts that if a monomial ideal is p-Borel-fixed, then its \(\mathbb{N}\)-graded Betti table, after passing to any field, does not depend on the field. More precisely, we show that, for any monomial ideal I in a polynomial ring S over the ring \(\mathbb{Z}\) of integers and for any prime number p, there is a p-Borel-fixed monomial S-ideal J such that a region of the multigraded Betti table of \(J(S \otimes_{\mathbb{Z}}\ell)\) is in one-to-one correspondence with the multigraded Betti table of \(I(S \otimes_{\mathbb{Z}}\ell)\) for all fields of arbitrary characteristic. There is no analogous statement for Borel-fixed ideals in characteristic zero.

Additionally, the construction also shows that there are p-Borel-fixed ideals with noncellular minimal resolutions.


Graded free resolutions Positive characteristic Borel-fixed ideals Cellular resolutions 



The work of the first author was supported by a grant from the Simons Foundation (209661 to G.C.). The second author was partially supported by a CMI Faculty Development Grant. In addition, both authors thank Mathematical Sciences Research Institute, Berkeley CA, where part of this work was done, for support and hospitality during Fall 2012.


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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsPurdue UniversityWest LafayetteUSA
  2. 2.Chennai Mathematical InstituteSiruseriIndia

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