# Presentations of rings with non-trivial semidualizing modules

@article{Jorgensen2009PresentationsOR, title={Presentations of rings with non-trivial semidualizing modules}, author={David A. Jorgensen and Graham J. Leuschke and Sean Sather-Wagstaff}, journal={Collectanea Mathematica}, year={2009}, volume={63}, pages={165-180} }

Let R be a commutative noetherian local ring. A finitely generated R-module C is semidualizing if it is self-orthogonal and $${{\rm Hom}_R(C,C)\cong R}$$ . We prove that a Cohen–Macaulay ring R with dualizing module D admits a semidualizing module C satisfying $${R\ncong C \ncong D}$$ if and only if it is a homomorphic image of a Gorenstein ring in which the defining ideal decomposes in a cohomologically independent way. This expands on a well-known result of Foxby, Reiten and Sharp saying that… Expand

#### 14 Citations

Presentations of rings with a chain of semidualizing modules

- Mathematics
- 2014

Inspired by Jorgensen et. al., it is proved that if a Cohen--Macaulay local ring $R$ with dualizing module admits a suitable chain of semidualizing $R$--modules of length $n$, then $R\cong… Expand

On Balance for Relative Homology

- Mathematics
- 2016

Let R be a ring. It is reasonable to use a strict 𝒲𝒳-resolution of a right R-module M of finite 𝒳-projective dimension, where 𝒳 denotes a subcategory of right R-modules closed under extensions… Expand

Chains of semidualizing modules

- Mathematics
- 2016

Let (R, 𝔪,k) be a commutative Noetherian local ring. We study the suitable chains of semidualizing R-modules. We prove that when R is Artinian, the existence of a suitable chain of semidualizing… Expand

Gorenstein Injective Filtrations Over Cohen-Macaulay Rings with Dualizing Modules

- Mathematics
- 2014

Over a noetherian ring, it is a classic result of Matlis that injective modules admit direct sum decompositions into injective hulls of quotients by prime ideals. We show that over a Cohen-Macaulay… Expand

Cotorsion dimensions relative to semidualizing modules

- Mathematics
- 2016

Let C be a semidualizing R-module, where R is a commutative ring. We first introduce the definition of C-cotorsion modules, and obtain the properties of C-cotorsion modules. As applications, we give… Expand

BALANCE FOR RELATIVE HOMOLOGY WITH RESPECT TO SEMIDUALIZING MODULES

- Mathematics
- 2015

Abstract. We derive in the paper the tensor product functor −⊗ R − byusing proper GP C -resolutions, where C is a semidualizing module. Aftergiving several cases in which diﬀerent relative homologies… Expand

Relative and Tate homology with respect to semidualizing modules

- Mathematics
- 2014

We introduce and investigate in this paper a kind of Tate homology of modules over a commutative coherent ring based on Tate ℱC-resolutions, where C is a semidualizing module. We show firstly that… Expand

Categorical and homological aspects of module theory over commutative rings

- Mathematics
- 2015

The purpose of this work is to understand the structure of the subcategories of mod(R) and the derived category D(R) for a commutative Noetherian ring R. Special focus is given to categories… Expand

Naive liftings of DG modules

- Mathematics
- 2021

Let n be a positive integer, and let A be a strongly commutative differential graded (DG) algebra over a commutative ring R. Assume that (a) B = A[X1, . . . ,Xn] is a polynomial extension of A, where… Expand

Remarks on torsionfreeness and its applications

- Mathematics
- 2014

ABSTRACT In this article, we shall characterize torsionfreeness of modules with respect to a semidualizing module in terms of the Serre’s condition (Sn). As its applications, we give a… Expand

#### References

SHOWING 1-10 OF 32 REFERENCES

The converse to a theorem of Sharp on Gorenstein modules

- Mathematics
- 1972

Let A be a commutative local Noetherian ring with identity of Krull dimension n, m its maximal ideal. Sharp has proved that if A is Cohen-Macauley and a homomorphic image of a Gorenstein local ring,… Expand

Bass numbers and semidualizing complexes

- Mathematics
- 2008

Let R be a commutative local Noetherian ring. We prove that the existence of a chain of semidualizing R-complexes of length .d C 1/ yields a degree-d polynomial lower bound for the Bass numbers of R.… Expand

Semi-dualizing modules and related Gorenstein homological dimensions

- Mathematics
- 2004

Abstract A semi-dualizing module over a commutative noetherian ring A is a finitely generated module C with RHom A ( C , C ) ≃ A in the derived category D ( A ) . We show how each such module gives… Expand

Ring homomorphisms and finite Gorenstein dimension

- Mathematics
- 1997

The local structure of homomorphisms of commutative noetherian rings is investigated from the point of view of dualizing complexes. A concept of finite Gorenstein dimension, which substantially… Expand

W-Gorenstein modules

- Mathematics
- 2011

Abstract Let W be a self-orthogonal class of left R-modules. We introduce and study W -Gorenstein modules as a common generalization of some known modules such as Gorenstein projective (injective)… Expand

Semi-dualizing complexes and their Auslander categories

- Mathematics
- 2001

Let R be a commutative Noetherian ring. We study R–modules, and complexes of such, with excellent duality properties. While their common properties are strong enough to admit a rich theory, we count… Expand

On modules with trivial self-extensions

- Mathematics
- 1988

Throughout this note, all rings will be self-basic connected artin algebras over a fixed commutative local artin ring k and all modules will be finitely generated. Homomorphisms will be written on… Expand

ABSOLUTE, RELATIVE, AND TATE COHOMOLOGY OF MODULES OF FINITE GORENSTEIN DIMENSION

- Mathematics
- 2002

We study finitely generated modules $M$ over a ring $R$, noetherian on both sides. If $M$ has finite Gorenstein dimension $\mbox{G-dim}_RM$ in the sense of Auslander and Bridger, then it determines… Expand

Tate cohomology with respect to semidualizing modules

- Mathematics
- 2009

We investigate Tate cohomology of modules over a commutative noetherian ring with respect to semidualizing modules. We identify classes of modules admitting Tate resolutions and analyze the… Expand

Concerning adjunctions to algebras

- Mathematics
- 1932

In §3 of his paper, Algebras which do not possess a finite basis ^ J. H. M. Wedderburn gives a set of postulates and definitions for an algebra. The question of the possibility of adjoining an… Expand