Betti tables of p-Borel-fixed ideals

الأصل · EN

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 -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 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 ⊗ ℓ) is in one-to-one correspondence with the multigraded Betti table of I(S ⊗ ℓ) for all fields ℓ 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.