Annihilator of Ext

TL;DR

This paper studies the ideal D(I) that annihilates Ext modules, exploring when it is contained within the integral closure of I, and develops structural properties and applications related to this containment.

Contribution

It establishes broad conditions for the containment D(I) ⊆ overline{I} and relates D(I) to various algebraic structures, providing new insights and examples.

Findings

Containment D(I) ⊆ overline{I} holds for specific classes of ideals.

Structural properties of D(I) relate it to symbolic powers and trace ideals.

Applications include criteria for triviality of reflexive modules and connections with local cohomology.

Abstract

We investigate the higher divisorial ideal $D(I):= Ann(Ext^g_R(R/I,R))$ associated to an ideal I of grade g. Our main focus is the containment problem $D(I) \subseteq \overline{I}$. We show 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 suitable homological conditions. Conversely, we construct explicit examples demonstrating the necessity of these 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 among annihilators of Ext, conductor ideals, and local cohomology.