Betti tables of $p$-Borel-fixed ideals

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 $p$-Borel-fixed, then its $\naturals$-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 $\ints$ 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 $J(S \otimes_\ints \ell)$ is in one-to-one correspondence with the multigraded Betti table of $I(S \otimes_\ints \ell)$ for all fields $\ell$ 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 non-cellular minimal resolutions.