On the Cohen-Macaulayness of the conormal module of an ideal

TL;DR

This paper explores the Cohen-Macaulayness of the conormal module of an ideal, providing positive results for certain classes and counterexamples in general, thereby addressing a long-standing conjecture in commutative algebra.

Contribution

It offers new conditions under which the Cohen-Macaulayness of the conormal module implies the Gorenstein property, and presents counterexamples showing the limits of this implication.

Findings

Positive results for specific classes of ideals.

Counterexamples demonstrating the failure of the conjecture in general.

Identification of Cohen-Macaulay ideals with non-Cohen-Macaulay squares.

Abstract

In the present paper we investigate a question stemming from a long-standing conjecture of Vasconcelos: given a generically a complete intersection perfect ideal I in a regular local ring R, is it true that if I/I^2 (or R/I^2) is Cohen-Macaulay then R/I is Gorenstein? Huneke and Ulrich, Minh and Trung, Trung and Tuan and - very recently - Rinaldo Terai and Yoshida, already considered this question and gave a positive answer for special classes of ideals. We give a positive answer for some classes of ideals, however, we also exhibit prime ideals in regular local rings and homogeneous level ideals in polynomial rings showing that in general the answer is negative. The homogeneous examples have been found thanks to the help of J. C. Migliore. Furthermore, the counterexamples show the sharpness of our main result. As a by-product, we exhibit several classes of Cohen-Macaulay ideals whose square is not Cohen-Macaulay. Our methods work both in the homogeneous and in the local settings.