On reduction numbers and Castelnuovo-Mumford regularity of blowup rings and modules

TL;DR

This paper explores the relationship between reduction numbers and Castelnuovo-Mumford regularity of blowup rings and modules, using Ratliff-Rush closure, with new results answering longstanding questions and generalizing previous theorems.

Contribution

It introduces new connections between reduction numbers and regularity, extends known results to higher dimensions, and provides applications to ideals of linear type and Ulrich ideals.

Findings

Answered specific cases of a question by Rossi, Trung, and Trung.

Generalized a result of Mafi to arbitrary dimension.

Provided characterizations of ideals of linear type and progress on Ulrich ideals.

Abstract

We prove new results on the connections between reduction numbers and the Castelnuovo-Mumford regularity of blowup algebras and blowup modules, the key basic tool being the operation of Ratliff-Rush closure. First, we answer in two particular cases a question of M. E. Rossi, D. T. Trung, and N. V. Trung about Rees algebras of ideals in two-dimensional Buchsbaum local rings, and we even ask whether one of such situations always holds. In another theorem we generalize a result of A. Mafi on ideals in two-dimensional Cohen-Macaulay local rings, by extending it to arbitrary dimension (and allowing for the setting relative to a Cohen-Macaulay module). We derive a number of applications, including a characterization of (polynomial) ideals of linear type, progress on the theory of generalized Ulrich ideals, and improvements of results by other authors.