Betti tables of p-Borel-fixed ideals |
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Authors: | Giulio Caviglia Manoj Kummini |
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Institution: | 1. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA 2. Chennai Mathematical Institute, Siruseri, Tamilnadu, 603103, India
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Abstract: | 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. |
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