On linear resolution of powers of an ideal

TL;DR

This paper extends existing bounds on the regularity of ideal powers and provides a criterion based on Rees algebra to determine when high powers have linear resolutions, with applications to specific ideals.

Contribution

It introduces a simple Rees algebra-based criterion for linear resolutions of high powers of ideals, generalizing previous results and including computational tools.

Findings

High powers of certain ideals have linear resolutions if and only if the power is not 2.

The criterion applies to important classes of ideals, such as J and J1.

The procedures are implemented in the CoCoA algebra system.

Abstract

In this paper we give a generalization of a result of Herzog, Hibi, and Zheng providing an upper bound for regularity of powers of an ideal. As the main result of the paper, we give a simple criterion in terms of Rees algebra of a given ideal to show that high enough powers of this ideal have linear resolution. We apply the criterion to two important ideals $J,J_{1}$ for which we show that $J^{k},$ and $J_{1}^{k}$ have linear resolution if and only if $k\neq 2.$ The procedures we include in this work is encoded in computer algebra package CoCoA.