Regularity jumps for powers of ideals

الأصل · EN

The Castelnuovo-Mumford regularity (I) is one of the most important invariants of a homogeneous ideal I in a polynomial ring. A basic question is how the regularity behaves with respect to taking powers of ideals. It is known that in the long-run (Iᵏ) is a linear function of k. We show that in the short-run the regularity of Iᵏ can be quite "irregular". For any given integer d>1 we construct an ideal J generated by d+5 monomials of degree d+1 in 4 variables such that (Jᵏ)=k(d+1) for every k<d and (Jᵈ)≥ d(d+1)+d-1.