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

Kaito Kimura

TL;DR

The paper addresses containment questions for conductor, parameter test ideals, $F$-ideals, and trace ideals in rings with finite phantom projective dimension, establishing non-containment in broad classes of quasi-Gorenstein complete local domains while providing Cohen–Macaulay counterexamples that exhibit containment. It develops a unified, derived-category framework based on dualizing complexes to extend and connect prior results (e.g., Dey–Dutta, Smith, Ikeda, Asgharzadeh) and derives concrete corollaries showing non-containment of $ au(oldsymbol{ au})$ in $Ioldsymbol{ au}$ and of $ ext{tr}_R(M)$ in $I$ under various hypotheses, including the quasi-Gorenstein case. The work demonstrates that the parameter test ideal and the conductor are not contained in ideals of finite phantom projective dimension when $R$ is a complete local domain, and it strengthens the monomial conjecture in this setting. Notably, the authors present dimension-two Cohen–Macaulay counterexamples illustrating that containment phenomena can occur outside CM and in minimal multiplicity cases, thereby answering Huneke–Swanson negatively and enriching the landscape of $F$-singularity behavior via a robust, modern derived-tools approach.

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.

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

TL;DR

The paper addresses containment questions for conductor, parameter test ideals, -ideals, and trace ideals in rings with finite phantom projective dimension, establishing non-containment in broad classes of quasi-Gorenstein complete local domains while providing Cohen–Macaulay counterexamples that exhibit containment. It develops a unified, derived-category framework based on dualizing complexes to extend and connect prior results (e.g., Dey–Dutta, Smith, Ikeda, Asgharzadeh) and derives concrete corollaries showing non-containment of in and of in under various hypotheses, including the quasi-Gorenstein case. The work demonstrates that the parameter test ideal and the conductor are not contained in ideals of finite phantom projective dimension when is a complete local domain, and it strengthens the monomial conjecture in this setting. Notably, the authors present dimension-two Cohen–Macaulay counterexamples illustrating that containment phenomena can occur outside CM and in minimal multiplicity cases, thereby answering Huneke–Swanson negatively and enriching the landscape of -singularity behavior via a robust, modern derived-tools approach.

Abstract

In this paper, we consider whether parameter test ideals, conductors, -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.
Paper Structure (3 sections, 14 theorems, 21 equations)

This paper contains 3 sections, 14 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Parameter test ideal and $F$-ideal
  3. Dualizing complex and standard conditions

Key Result

Theorem 1.3

Let $R$ be a local ring of prime characteristic $p$, $I$ a proper ideal of $R$ of finite projective dimension, and $J$ a non-zero $F$-ideal. If $R$ is either a Cohen--Macaulay ring or an excellent equidimensional reduced ring, then $J$ is not contained in $I$.

Theorems & Definitions (41)

  • Theorem 1.3: Theorem \ref{['F-ideal and fpd']}
  • Theorem 1.4: Corollary \ref{['nonCM original']}
  • Corollary 1.5: Corollary \ref{['nonCM cor']}
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • Remark 2.5
  • Corollary 2.6
  • ...and 31 more