Annihilator of Ext

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.