 

Alexandre Tchernev
On the grades of order ideals


Published: 
April 12, 2005

Keywords: 
Order ideals, syzygies, syzygy theorem 
Subject: 
13D02, 13D22 


Abstract
Let R be a commutative Noetherian local ring, let M be
a finitely generated Rmodule 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 O_{M}(u)={f(u)  f∈Hom_{R}(M,R)}.
The main result is that when M is a kth syzygy module and
pd_{R} M≦ 1 then grade_{R} O_{M}(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 CohenMacaulay.


Author information
Department of Mathematics, University at Albany, SUNY, Albany, NY 12222
tchernev@math.albany.edu
http://math.albany.edu:8000/~tchernev/

