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Annihilator of Ext

Mohsen Asgharzadeh

TL;DR

The paper investigates the higher divisorial annihilator $D(I)$ of an ideal $I$ via $D(I) = Ann(Ext^g_R(R/I,R))$ and the containment problem $D(I) \subseteq$ integral closure of $I$, developing a structural theory that places $D(I)$ between the unmixed part $I^{unm}$ and the radical of $I$. It proves broad positive containment results for classes such as unmixed ideals of finite projective dimension on 3-dimensional quasi-normal rings, parameter ideals in quasi-Gorenstein rings, and powers of perfect ideals under homological conditions, while also providing explicit counterexamples that illustrate limitations. The work connects $D(I)$ to trace ideals, Frobenius/tight closures, and conductor ideals, and derives applications to triviality criteria for reflexive modules and vector bundles on punctured spectra. It also maps intricate relations between annihilators of Ext, local cohomology, and integral closures, offering a rich homological-idealic framework with geometric implications.

Abstract

We investigate the higher divisorial ideal \( D(I) := \Ann\!\bigl(Ext^g_R(R/I,R)\bigr) \) associated to an ideal \(I\) of grade \(g\). Our main focus is the containment problem \( D(I) \subseteq \overline{I} \). We establish that this inclusion holds for broad classes of ideals, including unmixed ideals of finite projective dimension over 3-dimensional quasi-normal rings, parameter ideals in quasi-Gorenstein rings, and powers of perfect ideals under certain homological conditions. Conversely, we construct explicit examples showing the necessity of hypotheses. We develop structural properties of \(D(I)\), relating it to unmixed parts, reflexive closures, symbolic powers, Frobenius closure, and trace ideals. Applications include criteria for the triviality of reflexive modules and vector bundles on punctured spectra, as well as new connections between annihilators of Ext, conductor ideals, and local cohomology.

Annihilator of Ext

TL;DR

The paper investigates the higher divisorial annihilator of an ideal via and the containment problem integral closure of , developing a structural theory that places between the unmixed part and the radical of . It proves broad positive containment results for classes such as unmixed ideals of finite projective dimension on 3-dimensional quasi-normal rings, parameter ideals in quasi-Gorenstein rings, and powers of perfect ideals under homological conditions, while also providing explicit counterexamples that illustrate limitations. The work connects to trace ideals, Frobenius/tight closures, and conductor ideals, and derives applications to triviality criteria for reflexive modules and vector bundles on punctured spectra. It also maps intricate relations between annihilators of Ext, local cohomology, and integral closures, offering a rich homological-idealic framework with geometric implications.

Abstract

We investigate the higher divisorial ideal \( D(I) := \Ann\!\bigl(Ext^g_R(R/I,R)\bigr) \) associated to an ideal of grade . Our main focus is the containment problem \( D(I) \subseteq \overline{I} \). We establish that this inclusion holds for broad classes of ideals, including unmixed ideals of finite projective dimension over 3-dimensional quasi-normal rings, parameter ideals in quasi-Gorenstein rings, and powers of perfect ideals under certain homological conditions. Conversely, we construct explicit examples showing the necessity of hypotheses. We develop structural properties of \(D(I)\), relating it to unmixed parts, reflexive closures, symbolic powers, Frobenius closure, and trace ideals. Applications include criteria for the triviality of reflexive modules and vector bundles on punctured spectra, as well as new connections between annihilators of Ext, conductor ideals, and local cohomology.
Paper Structure (7 sections, 27 theorems, 72 equations)

This paper contains 7 sections, 27 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and structural concepts
  3. Higher divisorial
  4. More on ideals of finite projective dimension
  5. Connections to trace
  6. Low dimension and small height
  7. Connections to triviality of vector bundles

Key Result

Proposition 2.8

Let $R$ be universally catenary, quasi-Gorenstein and $I$ be a parameter ideal of $g:=\operatorname{codim}(I)$. Then $\operatorname{Ann}(\operatorname{Ext}^g(R/I, R)) \subseteq \overline{I}.$

Theorems & Definitions (77)

  • proof
  • Remark 2.2
  • proof
  • Example 2.4
  • Conjecture 2.6
  • Remark 2.7
  • Proposition 2.8
  • proof
  • Corollary 2.9
  • Remark 2.10
  • ...and 67 more