Presentations of rings with a chain of semidualizing modules

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 $R\cong Q/(I_1+\cdots +I_n)$ for some Gorenstein ring $Q$ and ideals $I_1,\dots , I_n$ of $Q$; and, for each $\Lambda \subseteq [n]$, the ring $Q/(\sum _{\ell \in \Lambda } I_\ell )$ has some interesting cohomological properties. This extends the result of Jorgensen et al., and also of Foxby and Reiten.

Relative and Tate homology with respect to semidualizing modules

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 the class of modules admitting a Tate ℱC-resolution is equal to the class of modules of finite 𝒢(ℱC)-projective dimension. Then an Avramov–Martsinkovsky type exact sequence is constructed to connect such Tate homology functors and relative homology functors. Finally, motivated by the idea of Sather–Wagstaff et al. [Comparison of relative cohomology theories with respect to semidualizing modules, Math. Z. 264 (2010) 571–600], we establish a balance result for such Tate homology over a Cohen–Macaulay ring with a dualizing module by using a good conclusion provided in [E. E. Enochs, S. E. Estrada and A. C. Iacob, Balance with unbounded complexes, Bull. London Math. Soc. 44 (2012) 439–442].

DUALIZING MODULES AND n-PERFECT RINGS

In this article we extend the results about Gorenstein modules and Foxby duality to a non-commutative setting. This is done in §3 of the paper, where we characterize the Auslander and Bass classes which arise whenever we have a dualizing module associated with a pair of rings. In this situation it is known that flat modules have finite projective dimension. Since this property of a ring is of interest in its own right, we devote §2 of the paper to a consideration of such rings. Finally, in the paper’s final section, we consider a natural generalization of the notions of Gorenstein modules which arises when we are in the situation of §3, i.e. when we have a dualizing module.AMS 2000 Mathematics subject classification: Primary 16D20

A noncommutative generalization of Auslander's last theorem

We show that every finitely generated leftR-module in the Auslander class over ann-perfect ringRhaving a dualizing module and admitting a Matlis dualizing module has a Gorenstein projective cover.

Foxby Equivalence and Cotorsion Theories relative to Semi-dualizing Modules

Foxby duality has proven to be an important tool in studying the category of modules over a local Cohen-Macaulay ring admitting a dualizing module. Recently the notion of a semi-dualizing module has been given [2]. Given a semi-dualizing module the relative Foxby classes can be defined and there is still an associated Foxby duality. We consider these classes (separately called the Auslander and Bass classes) and two naturally defined subclasses which are equivalent to the full subcategories of injective and flat modules. We consider the question of when these subclasses form part of one of the two classes of a cotorsion theory. We show that when this is the case, the associated cotorsion theory is not only complete but in fact is perfect. We show by examples that even when the semi-dualizing module is in fact dualizing over a local Cohen-Macaulay ring it both may or may not occur that we get this associated cotorsion theory.

On Matlis dualizing modules

We consider rings admitting a Matlis dualizing moduleE. We argue that ifRadmits two such dualizing modules, then a module is reflexive with respect to one if and only if it is reflexive with respect to the other. Using this fact we argue that the number (whether finite or infinite) of distinct dualizing modules equals the number of distinct invertible(R,R)-bimodules. We show by example that this number can be greater than one.