Cohen-Macaulay fiber cones and defining ideal of Rees algebras of modules

TL;DR

This paper explores the use of generic Bourbaki ideals to analyze the Cohen-Macaulay property of fiber cones and the defining ideals of Rees algebras of modules, extending previous methods through a new deformation condition.

Contribution

It introduces a new deformation condition that broadens the applicability of generic Bourbaki ideals in studying Rees algebras and fiber cones of modules.

Findings

Established a deformation criterion linking Rees algebras of modules and their Bourbaki ideals.

Extended the use of generic Bourbaki ideals beyond previous limitations.

Provided new insights into the Cohen-Macaulay property of fiber cones.

Abstract

Generic Bourbaki ideals were introduced by Simis, Ulrich and Vasconcelos to study the Cohen-Macaulay property of Rees algebras of modules. In this article we prove that the same technique can sometimes be used to investigate the Cohen-Macaulay property of fiber cones of modules and to study the defining ideal of Rees algebras. This is possible as long as the Rees algebra of a given module $E$ is a deformation of the Rees algebra of a generic Bourbaki ideal $I$ of $E$. Our main technical result provides a deformation condition that in fact extends the applicability of generic Bourbaki ideals to situations not covered by previous work.