When are the rings $I:I$ Gorenstein?

TL;DR

This paper investigates conditions under which the endomorphism ring of an ideal in a Noetherian local ring is Gorenstein, linking it to the Gorenstein property of certain Rees algebras, with illustrative examples.

Contribution

It establishes new criteria for the Gorensteinness of the ideal endomorphism ring in relation to Rees algebras, extending previous work on parameter ideals.

Findings

Characterization of when $I:I$ is Gorenstein.

Connection between $I:I$ Gorensteinness and Rees algebra Gorensteinness.

Examples illustrating the theoretical results.

Abstract

Let $I ~(\ne A)$ be an ideal of a $d$-dimensional Noetherian local ring $A$ with $\operatorname{ht}_AI \ge 2$, containing a non-zerodivisor. The problem of when the ring $I:I=\operatorname{End}_AI$ is Gorenstein is studied, in connection with the problem of the Gorensteinness in Rees algebras $\mathcal{R}_A(Q^d)$ for certain parameter ideals $Q$ of $A$, that was closely explored by the preceding paper of the second and third authors. Examples are given.