Ideals with larger projective dimension and regularity

Abstract

We define a family of homogeneous ideals with large projective dimension and regularity relative to the number of generators and their common degree. This family subsumes and improves upon constructions given in [Cav04] and [McC]. In particular, we describe a family of three-generated homogeneous ideals in arbitrary characteristic whose projective dimension grows asymptotically as √ d √ d−1 .

Bounding Projective Dimension

“…It is also noted in [4] that some of the ideals I g,(m1,m2,...,mn) have regularity larger than d 2 − 2. It would be interesting to compute the regularity of this new family of ideals as this would give insight into the regularity version of Stillman's Question.…”

“…This construction can be considered as an inductive version of the family in the previous section. The family was constructed in joint work by the two authors along with Beder, Núñez-Betancourt, Snapp and Stone in [4]. Fix integers g ≥ 2 and a tuple of integers m 1 , .…”

A tight bound on the projective dimension of four quadrics

“…al. [5] show that the projective dimension of ideals generated by 3 degree-d forms can grow exponentially with respect to d. While (1) above shows that pd(S/I) is bounded for any ideal generated by N quadrics, these bounds are exponential in N . It is clear already in the case of an ideal generated by 3 quadrics that these bounds are far from optimal (cf.…”

On the maximal graded shifts of ideals and modules

“…Examples derived from the Mayr-Meyer ideals [15] show that this upper bound is nearly optimal in that families of ideals generated by quadrics in n variables can have first syzygies whose degree grows doubly exponentially in n. Other examples of ideals with large regularity (e.g. [3], [4], [5]) also have large degree first syzygies. Later examples of "designer ideals" constructed by Ullery [19] showed that for any strictly increasing sequence of positive integers 2 ≤ a 1 < a 2 < · · · < a m , there is an ideal I in a polynomial ring S such that T i (S/I) = a i for all 1 ≤ i ≤ m. However, these ideals also satisfy T m+i (S/I) = a m + i for i > 0 (i.e.…”