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In the present paper we investigate a question stemming from a long-standingconjecture of Vasconcelos: given a generically a complete intersection perfectideal I in a regular local ring R, is it true that if I/I^2 (or R/I^2) isCohen-Macaulay then R/I is Gorenstein? Huneke and Ulrich, Minh and Trung, Trungand Tuan and - very recently - Rinaldo Terai and Yoshida, already consideredthis question and gave a positive answer for special classes of ideals. We givea positive answer for some classes of ideals, however, we also exhibit primeideals in regular local rings and homogeneous level ideals in polynomial ringsshowing that in general the answer is negative. The homogeneous examples havebeen found thanks to the help of J. C. Migliore. Furthermore, thecounterexamples show the sharpness of our main result. As a by-product, weexhibit several classes of Cohen-Macaulay ideals whose square is notCohen-Macaulay. Our methods work both in the homogeneous and in the localsettings.

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