Ideals with Larger Projective Dimension and Regularity

TL;DR

This paper introduces a new family of homogeneous ideals with significantly larger projective dimension and regularity, surpassing previous constructions, and provides explicit examples with asymptotic growth in projective dimension.

Contribution

It constructs a novel family of ideals with larger projective dimension and regularity, improving upon prior examples and applicable in arbitrary characteristic.

Findings

Family of ideals with large projective dimension and regularity

Asymptotic growth of projective dimension as sqrt{d}^(sqrt(d)-1)

Applicable in arbitrary characteristic

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 sqrt{d}^(sqrt(d) - 1).