New York Journal of Mathematics
Volume 11 (2005) 151-156


Alexandre Tchernev

On the grades of order ideals

Published: April 12, 2005
Keywords: Order ideals, syzygies, syzygy theorem
Subject: 13D02, 13D22

Let R be a commutative Noetherian local ring, let M be a finitely generated R-module of finite projective dimension, and let u∈ M be a minimal generator of M. We investigate in a characteristic free setting the grade of the order ideal OM(u)={f(u) | f∈HomR(M,R)}. The main result is that when M is a k-th syzygy module and pdR M≦ 1 then gradeR OM(u)≧ k; in particular if M is an ideal of projective dimension at most 1 then every minimal generator of M is a regular element of R. As an application we show that the minimal generators of M are regular elements of R also in the case when M is a Gorenstein ideal of grade 3, in the case when M is a three generated ideal, and in the case when M is an almost complete intersection ideal of grade 3 and R is Cohen-Macaulay.

Author information

Department of Mathematics, University at Albany, SUNY, Albany, NY 12222