Geodesic
Betti tables of $p$-Borel-fixed ideals
Journal of Algebraic Combinatorics, Tome 39 (2014) no. 3, pp. 711-718.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

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 $\ell$ 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 noncellular minimal resolutions.
Classification : 13D02, 13F20
Keywords: Borel-fixed ideals, graded free resolutions, cellular resolutions 
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Caviglia, Giulio; Kummini, Manoj. Betti tables of $p$-Borel-fixed ideals. Journal of Algebraic Combinatorics, Tome 39 (2014) no. 3, pp. 711-718. http://geodesic.mathdoc.fr/item/JAC_2014__39_3_a2/