Characteristic-free bounds for the Castelnuovo-Mumford regularity

TL;DR

This paper establishes bounds for the Castelnuovo-Mumford regularity of homogeneous ideals that are independent of characteristic, extending known results from characteristic zero to positive characteristic.

Contribution

It provides a characteristic-free proof of the upper bound on regularity, confirming a question posed by Bayer and Mumford, and introduces a new argument using Bayer and Stillman's criterion.

Findings

The known upper bound in characteristic zero also holds in positive characteristic.

Analysis of Giusti's proof offers insights into combinatorial properties involved.

A new proof employs Bayer and Stillman's criterion for regularity detection.

Abstract

We study bounds for the Castelnuovo-Mumford regularity of homogeneous ideals in a polynomial ring in terms of the number of variables and the degree of the generators. In particular our aim is to give a positive answer to a question posed by Bayer and Mumford, by showing that the known upper bound in characteristic zero holds true also in positive characteristic. We first analyze Giusti's proof, which provides the result in characteristic 0, giving some insight on the combinatorial properties needed in that context. For the general case we provide a new argument which employs Bayer and Stillman criterion for detecting regularity.