Linear syzygies, flag complexes, and regularity

TL;DR

This paper constructs monomial ideals with degree-two generators, linear syzygies, and arbitrarily large regularity, revealing new insights into the growth of regularity in such ideals.

Contribution

It demonstrates the existence of monomial ideals with linear syzygies and unbounded regularity, and analyzes the typical growth rate of regularity in these ideals.

Findings

Existence of monomial ideals with arbitrary regularity R

Regularity of Gorenstein ideals is at most four

Regularity grows at most doubly logarithmically in most cases

Abstract

We show that for every positive integer R there exist monomial ideals generated in degree two, with linear syzygies, and regularity of the quotient equal to R. Such examples can not be found among Gorenstein ideals since the regularity of their quotients is at most four. We also show that for most monomial ideals generated in degree two and with linear syzygies the regularity grows at most doubly logarithmically in the number of variables.