Annihilator of $Ex{t}_R^g(R/I,R)$

Abstract

We investigate the higher divisorial ideal $D(I):=Ann(Ex{t}_R^g(R/I,R))$ 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. If $R$ is Cohen--Macaulay local, $I$ is height-unmixed, and suppose $pdi{m}_R(R/I)<\infty$, then we show $D_R(I)=I\subseteq \overline{I}.$ 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 the rigidity property of homological annihilators, a 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.