32 research outputs found
On the notion of Cohen-Macaulayness for non Noetherian rings
There exist many characterizations of Noetherian Cohen-Macaulay rings in the
literature. These characterizations do not remain equivalent if we drop the
Noetherian assumption. The aim of this paper is to provide some comparisons
between some of these characterizations in non Noetherian case. Toward solving
a conjecture posed by Glaz, we give a generalization of the Hochster-Eagon
result on Cohen-Macaulayness of invariant rings, in the context of non
Noetherian rings.Comment: 2
Gorenstein homological dimensions and Auslander categories
In this paper, we study Gorenstein injective, projective, and flat modules
over a Noetherian ring . For an -module , we denote by
and the Gorenstein projective and flat dimensions of ,
respectively. We show that if and only if provided the Krull dimension of is finite. Moreover, in the
case that is local, we correspond to a dualizing complex of
, the classes and of -modules. For a module
over a local ring , we show that if and only if or equivalently . In dual situation by
using the class , we provide a characterization of Gorenstein injective
modules.Comment: 15 page
The finiteness dimension of local cohomology modules and its dual notion
Let \fa be an ideal of a commutative Noetherian ring R and M a finitely
generated R-module. We explore the behavior of the two notions f_{\fa}(M), the
finiteness dimension of M with respect to \fa, and, its dual notion q_{\fa}(M),
the Artinianess dimension of M with respect to \fa. When (R,\fm) is local and
r:=f_{\fa}(M) is less than f_{\fa}^{\fm}(M), the \fm-finiteness dimension of M
relative to \fa, we prove that H^r_{\fa}(M) is not Artinian, and so the filter
depth of \fa on M doesn't exceeds f_{\fa}(M). Also, we show that if M has
finite dimension and H^i_{\fa}(M) is Artinian for all i>t, where t is a given
positive integer, then H^t_{\fa}(M)/\fa H^t_{\fa}(M) is Artinian. It
immediately implies that if q:=q_{\fa}(M)>0, then H^q_{\fa}(M) is not finitely
generated, and so f_{\fa}(M)\leq q_{\fa}(M).Comment: 14 pages, to appear in Journal of Pure and Applied Algebr