Trace ideals, conductors, and ideals of finite (phantom) projective dimension

TL;DR

This paper investigates the relationship between certain special ideals and rings with finite phantom projective dimension, showing that some expected inclusions do not hold in specific classes of rings, and providing counterexamples.

Contribution

It demonstrates that parameter test ideals, conductors, and trace ideals are not contained in ideals with finite phantom projective dimension in quasi-Gorenstein complete local domains, answering a question of Huneke-Swanson.

Findings

Inclusions do not exist in quasi-Gorenstein complete local domains.

Counterexamples are provided in Cohen-Macaulay local rings.

The paper clarifies the limitations of certain ideal containments in rings with finite phantom projective dimension.

Abstract

In this paper, we consider whether parameter test ideals, conductors, $F$-ideals, and trace ideals are contained in an ideal whose quotient ring has finite phantom projective dimension (for example, ideals generated by a system of parameters or ideals with finite projective dimension). One of the main results asserts that such inclusions do not exist in quasi-Gorenstein complete local domains. We also provide examples of Cohen-Macaulay local rings with good properties where such inclusions occur, thus answering negatively a question of Huneke-Swanson.