Castelnuovo-Mumford regularity and powers

TL;DR

This paper introduces Castelnuovo-Mumford regularity for graded rings, provides a simple proof of the linearity of regularity of ideal powers, and characterizes ideals with linear resolutions via Rees rings.

Contribution

It offers a concise introduction to regularity over general rings, a simplified proof of a classical linearity result, and a new characterization of ideals with linear resolutions.

Findings

Regularity of powers of homogeneous ideals is eventually linear.

A simple proof of the classical linearity theorem is provided.

Characterization of ideals with linear resolutions using Rees rings.

Abstract

This note has two goals. The first is to give a short and self contained introduction to the Castelnuovo-Mumford regularity for standard graded ring $R$ over a general base ring. The second is to present a simple and concise proof of a classical result due to Cutkosky, Herzog and Trung and, independently, to Kodiyalam asserting that the regularity of powers of an homogeneous ideal $I$ of $R$ is eventually a linear function in $v$. Finally we show how the flexibility of the definition of the Castelnuovo-Mumford regularity over general base rings can be used to give a simple characterization of the ideals whose powers have a linear resolution in terms of the regularity of the Rees ring.