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.