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

JOURNAL OF ALGEBRA

卷:372

Article

Mantero, Paolo¹  Xie, Yu² 

[1] Purdue Univ, Dept Math, W Lafayette, IN 47906 USA

[2] Univ Notre Dame, Dept Math, Notre Dame, IN 46556 USA

关键词: Cohen-Macaulay;    Gorenstein;    Multiplicity;    Hilbert function;    Conormal module;   

DOI  :  10.1016/j.jalgebra.2012.07.039

来源: Elsevier

【 摘 要 】

In the present paper we investigate a question stemming from a long-standing conjecture of Vasconcelos: given a generically complete intersection perfect ideal I in a regular local ring R, when is it true that the Cohen-Macaulayness of I/I-2 (or R/I-2) implies that R/I is Gorenstein? This property is known to hold for licci ideals and, essentially, squarefree monomial ideals. We show that a positive answer actually holds for every monomial ideal. We then give a positive answer for several special classes of ideals and provide application to algebroid curves with low multiplicity. We also exhibit prime ideals in regular local rings and homogeneous level ideals in polynomial rings for which the answer is negative and use them to show the sharpness of our main result, as they lie in the first class of ideals not covered by it. The homogeneous examples have been found thanks to the help of J.C. Migliore. 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. Published by Elsevier Inc.

【 授权许可】

Free