[Paper Review] Trace of canonical modules, annihilator of Ext, and classes of rings close to being Gorenstein

Hailong Dao, Toshinori Kobayashi+1 more|arXiv (Cornell University)|May 5, 2020

Commutative Algebra and Its Applications27 references4 citations

TL;DR

This paper establishes new characterizations of trace ideals of canonical modules in terms of annihilators of Ext modules, particularly linking them to Gorenstein-like properties. It introduces weakly almost Gorenstein rings—generalizing almost Gorenstein rings in dimension one—and proves that nearly Gorenstein rings are preserved under reduction by regular sequences under certain conditions, offering a new version of the Tachikawa conjecture for type 2 rings.

ABSTRACT

In this note we study trace ideals of canonical modules. Characterizations of the trace ideals in terms of annihilators of certain Ext modules are given. We apply our results to study many classes of rings close to being Gorenstein that appear in recent literature. We discuss the behavior of nearly Gorensteinness, introduced by Herzog-Hibi-Stamate, under reductions by regular sequences. A nearly Gorenstein version of the Tachikawa conjecture is posed, and an affirmative answer given for type 2 rings. We also introduce the class of weakly almost Gorenstein rings using the canonical module, and show that they are almost Gorenstein rings in dimension one, and are closely related to Teter rings in the artinian case. We explain their basic properties and give some examples.

Motivation & Objective

• To characterize the trace ideal of the canonical module using annihilators of Ext modules.
• To study rings close to being Gorenstein, especially nearly Gorenstein and almost Gorenstein rings.
• To investigate how nearly Gorenstein and weakly almost Gorenstein properties behave under reduction by regular sequences.
• To propose and partially resolve a nearly Gorenstein version of the Tachikawa conjecture.
• To explore connections between weakly almost Gorenstein rings and Teter rings in the Artinian case.

Proposed method

• Use trace ideal and annihilator techniques to relate Ext modules to canonical modules in Cohen–Macaulay local rings.
• Apply the transpose of the canonical module to derive new annihilator identities in the Gorenstein-on-punctured-spectrum case.
• Introduce the class of weakly almost Gorenstein rings via exact sequences involving the canonical module and residue field.
• Use the structure of socle and canonical duals to characterize weakly almost Gorenstein Artinian rings.
• Construct examples via quotients of power series rings by ideals satisfying $\mathfrak{n}I \subseteq J \subseteq I$ with $S/I$ Gorenstein.
• Leverage results on Burch ideals and Koszul homology to verify weakly almost Gorenstein properties in specific cases.

Experimental results

Research questions

• RQ1 How can the trace ideal of the canonical module be characterized via annihilators of Ext modules?
• RQ2 Under what conditions is the nearly Gorenstein property preserved under quotient by regular sequences?
• RQ3 What is the relationship between weakly almost Gorenstein rings and almost Gorenstein rings in dimension one?
• RQ4 How do weakly almost Gorenstein Artinian rings relate to Teter rings and self-canonical dual ideals?
• RQ5 Can a nearly Gorenstein version of the Tachikawa conjecture be formulated and verified for rings of type 2?

Key findings

• The trace ideal of the canonical module equals the annihilator of $\operatorname{Ext}^{>0}_R(\omega, \operatorname{mod}R)$ and $\operatorname{Ext}^1_R(\operatorname{CM}(R), R)$ in a Cohen–Macaulay local ring with a canonical module.
• If $R$ is Gorenstein on the punctured spectrum, then $\operatorname{tr}\omega = \operatorname{ann}\operatorname{Ext}^{d+1}_R(\operatorname{Tr}\omega, R)$, where $\operatorname{Tr}\omega$ is the transpose of $\omega$.
• A Cohen–Macaulay local ring is nearly Gorenstein if and only if $R/(\underline{x})$ is nearly Gorenstein for any regular sequence $\underline{x}$, or for some regular sequence in $\mathfrak{m}^2$ or in $\operatorname{tr}\omega$.
• In dimension one, almost Gorenstein rings coincide with weakly almost Gorenstein rings, and the property is preserved under quotient by regular elements in $\mathfrak{m}$ or $\mathfrak{m}^2$.
• For Artinian rings, weakly almost Gorenstein is equivalent to $R/\operatorname{soc}R$ being Teter, or having a self-canonical dual maximal ideal.
• Many weakly almost Gorenstein rings arise as $S/J$ where $S$ is a power series ring, $I \subseteq \mathfrak{n}^2$, $S/I$ is Gorenstein, and $\mathfrak{n}I \subseteq J \subseteq I$.