Betti tables of $p$-Borel-fixed ideals
Giulio Caviglia, Manoj Kummini

TL;DR
This paper disproves a conjecture that p-Borel-fixed monomial ideals have Betti tables independent of the field, showing field dependence and the existence of non-cellular minimal resolutions.
Contribution
It provides a counter-example to Pardue's conjecture, demonstrating field dependence of Betti tables for p-Borel-fixed ideals and the existence of non-cellular minimal resolutions.
Findings
Counter-example to Pardue's conjecture.
Betti tables of p-Borel-fixed ideals depend on the field.
Existence of p-Borel-fixed ideals with non-cellular minimal resolutions.
Abstract
In this note we provide a counter-example to a conjecture of K. Pardue [Thesis, Brandeis University, 1994.], which asserts that if a monomial ideal is -Borel-fixed, then its -graded Betti table, after passing to any field does not depend on the field. More precisely, we show that, for any monomial ideal in a polynomial ring over the ring of integers and for any prime number , there is a -Borel-fixed monomial -ideal such that a region of the multigraded Betti table of is in one-to-one correspondence with the multigraded Betti table of 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 -Borel-fixed ideals with non-cellular minimal resolutions.
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