Castelnuovo-Mumford regularity and Ratliff-Rush closure

TL;DR

This paper explores the connections between Castelnuovo-Mumford regularity and Ratliff-Rush closure, providing new insights into their computation, stability, and invariance, especially for monomial ideals in two variables.

Contribution

It establishes strong relationships between the regularity and Ratliff-Rush closure, confirming a conjecture for certain monomial ideals and enhancing understanding of their algebraic properties.

Findings

Regularity of Rees algebra and fiber ring are equal for large classes of monomial ideals in two variables.

Results improve methods for computing Ratliff-Rush closure and analyzing stability.

Confirmed Eisenbud and Ulrich's conjecture in specific cases.

Abstract

We establish strong relationships between the Castelnuovo-Mumford regularity and the Ratliff-Rush closure of an ideal. Our results have several interesting consequences on the computation of the Ratliff-Rush closure, the stability of the Ratliff-Rush filtration, the invariance of the reduction number, and the computation of the Castelnuovo-Mumford regularity of the Rees algebra and the fiber ring. In particular, we prove that the Castelnuovo-Mumford regularity of the Rees algebra and of the fiber ring are equal for large classes of monomial ideals in two variables, thereby verifying a conjecture of Eisenbud and Ulrich for these cases.