Betti tables of $p$-Borel-fixed ideals

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 $\text{naturals}$-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 $\text{ints}$ 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 $J(S{\otimes }_{\text{ints}}\mathrm{\ell })$ is in one-to-one correspondence with the multigraded Betti table of $I(S \otimes_\ints \ell)$forallfields$\ell$ 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.