Castelnuovo-Mumford regularity up to symmetry

TL;DR

This paper investigates the growth of Castelnuovo-Mumford regularity for symmetric ideals in large polynomial rings, establishing linear bounds and confirming conjectures in specific cases like Artinian and squarefree monomial ideals.

Contribution

It provides a linear upper bound for the regularity of symmetric ideals and verifies the conjecture of linear growth in key cases, advancing understanding of asymptotic regularity behavior.

Findings

Linear upper bound for regularity of symmetric ideals

Confirmation of linear growth conjecture in Artinian cases

Verification of conjecture for squarefree monomial ideals

Abstract

We study the asymptotic behavior of the Castelnuovo-Mumford regularity along chains of graded ideals in increasingly larger polynomial rings that are invariant under the action of symmetric groups. A linear upper bound for the regularity of such ideals is established. We conjecture that their regularity grows eventually precisely linearly. We establish this conjecture in several cases, most notably when the ideals are Artinian or squarefree monomial.