Sally Modules and Reduction Numbers of Ideals

TL;DR

This paper explores the relationship between the reduction number of primary ideals and the multiplicity of Sally modules, extending known results to more general rings and analyzing their structural properties.

Contribution

It develops a change of rings technique for Sally modules, connects Sally module structure to Cohen-Macaulayness, and extends results to Buchsbaum rings.

Findings

Extended Sally module results to non-Cohen-Macaulay rings.

Linked Sally module structure to Cohen-Macaulayness of the fiber ring.

Provided explicit realization of S_2-fication of Buchsbaum rings.

Abstract

We study the relationship between the reduction number of a primary ideal of a local ring relative to one of its minimal reductions and the multiplicity of the corresponding Sally module. This paper is focused on three goals: (i) To develop a change of rings technique for the Sally module of an ideal to allow extension of results from Cohen-Macaulay rings to more general rings. (ii) To use the fiber of the Sally modules of almost complete intersection ideals to connect its structure to the Cohen-Macaulayness of the special fiber ring. (iii) To extend some of the results of (i) to two-dimensional Buchsbaum rings. Along the way we provide an explicit realization of the S_2-fication of arbitrary Buchsbaum rings.