The Sally Modules of Ideals: A Survey

TL;DR

This survey explores Sally modules of ideals, focusing on their role in understanding the cohomological properties and depth relations between Rees algebras and associated graded rings, especially in higher dimensions.

Contribution

It provides a comprehensive overview of Sally modules, extending their application from finite colength filtrations to higher-dimensional filtrations.

Findings

Sally modules serve as a bridge for cohomological information.

They help analyze depth and other properties of Rees algebras.

The survey covers developments in higher-dimensional filtrations.

Abstract

The Sally module of a Rees algebra $\BB$ relative to one of its Rees subalgebras $\AA$ is a construct that can be used as a mediator for the trade-off of cohomological (e.g. depth) information between $\BB$ and the corresponding associated graded ring for several types of filtrations. While originally devised to deal with filtrations of finite colength, here we treat aspects of these developments for filtrations in higher dimensions as well.