A finite module M over a noetherian local ring R is said to be Gorenstein if Ext(k, M) = 0 for all i 6= dimR. A endomorphism φ : R → R of rings is called contracting if φ(m) ⊆ m for some i ≥ 1. Letting R denote the R-module R with action induced by φ, we prove: A finite R-module M is Gorenstein if and only if HomR( R,M) ∼= M and ExtiR( R,M) = 0 for 1 ≤ i ≤ depthR.

Let $(R,fm)$ be a Gorenstein local ring and$M,N$ be two finitely generated modules over $R$. Nagel proved that if $M$ and $N$ are inthe same even liaison class, thenone has $H^i_{fm}(M)cong H^i_{fm}(N)$ for all $iIn this paper, we provide a short proof to this result.

We propose a concept of module liaison that extends Gorenstein liaison of ideals and provides an equivalence relation among unmixed modules over a commutative Gorenstein ring. Analyzing the resulting equivalence classes we show that several results known for Gorenstein liaison are still true in the more general case of module liaison. In particular, we construct two maps from the set of even li...

The Gorenstein projective modules are proved to form a precovering class in the module category of a ring which has a dualizing complex. 0. Introduction This paper proves over a wide class of rings that the Gorenstein projective modules form a precovering class in the module category. Let me explain this statement. There are two terms of mystery, “Gorenstein projective modules” and “precovering...

We study Gorenstein dimension and grade of a module M over a filtered ring whose assosiated graded ring is a commutative Noetherian ring. An equality or an inequality between these invariants of a filtered module and its associated graded module is the most valuable property for an investigation of filtered rings. We prove an inequality G-dimM ≤ G-dimgrM and an equality gradeM = grade grM , whe...

It was proved by Beligiannis and Krause that over certain Artin algebras, there are Gorenstein flat modules which are not direct limits of finitely generated Gorenstein projective modules. That is, these algebras have no Gorenstein analogue of the Govorov-Lazard Theorem. We show that, in fact, there is a large class of rings without such an analogue. Namely, let R be a commutative local noether...

We prove versions of results of Foxby and Holm about modules of finite (Gorenstein) injective dimension and finite (Gorenstein) projective dimension with respect to a semidualizing module. We also verify two special cases of a question of Takahashi and White.

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...

A main ingredient for Kustin–Miller unprojection, as developed in [PR], is the module HomR(I, ωR), where R is a local Gorenstein ring and I a codimension one ideal with R/I Gorenstein. We prove a method of calculating it in a relative setting using resolutions. We give three applications. In the first we generalise a result of [CFHR]. The second and the third are about Tom and Jerry, two famili...