The Existence of Finitely Generated Modules of Finite Gorenstein Injective Dimension
نویسنده
چکیده
In this note, we study commutative Noetherian local rings having finitely generated modules of finite Gorenstein injective dimension. In particular, we consider whether such rings are Cohen-Macaulay.
منابع مشابه
Gorenstein Injective Dimensions and Cohen-Macaulayness
Let (R,m) be a commutative noetherian local ring. In this paper we investigate the existence of a finitely generated R-module of finite Gorenstein dimension when R is Cohen-Macaulay. We study the Gorenstein injective dimension of local cohomology of complexes and next we show that if R is a non-Artinian Cohen-Macaulay ring, which does not have the minimal multiplicity, then R has a finite gener...
متن کاملFoxby Duality and Gorenstein Injective and Projective Modules
In 1966, Auslander introduced the notion of the G-dimension of a finitely generated module over a Cohen-Macaulay noetherian ring and found the basic properties of these dimensions. His results were valid over a local Cohen-Macaulay ring admitting a dualizing module (also see Auslander and Bridger (Mem. Amer. Math. Soc., vol. 94, 1969)). Enochs and Jenda attempted to dualize the notion of G-dime...
متن کاملThe existence totally reflexive covers
Let $R$ be a commutative Noetherian ring. We prove that over a local ring $R$ every finitely generated $R$-module $M$ of finite Gorenstein projective dimension has a Gorenstein projective cover$varphi:C rightarrow M$ such that $C$ is finitely generated and the projective dimension of $Kervarphi$ is finite and $varphi$ is surjective.
متن کاملGorenstein flat and Gorenstein injective dimensions of simple modules
Let R be a right GF-closed ring with finite left and right Gorenstein global dimension. We prove that if I is an ideal of R such that R/I is a semi-simple ring, then the Gorensntein flat dimensnion of R/I as a right R-module and the Gorensntein injective dimensnnion of R/I as a left R-module are identical. In particular, we show that for a simple module S over a commutative Gorensntein ring R, ...
متن کاملGorenstein injective dimension, Bass formula and Gorenstein rings
Let (R,m, k) be a noetherian local ring. It is well-known that R is regular if and only if the injective dimension of k is finite. In this paper it is shown that R is Gorenstein if and only if the Gorenstein injective dimension of k is finite. On the other hand a generalized version of the so-called Bass formula is proved for finitely generated modules of finite Gorenstein injective dimension. ...
متن کامل