Construction of Modules with Finite Homological Dimensions

A new homological dimension, called G^*-dimension, is defined for every finitely generated module M over a local noetherian ring R. It is modeled on the CI-dimension of Avramov, Gasharov, and Peeva and has parallel properties. In particular, a ring R is Gorenstein if and only if every finitely generated R-module has finite G^*-dimension. The G^*-dimension lies between the CI-dimension and the G-dimension of Auslander and Bridger. This relation belongs to a longer sequence of inequalities, where a strict inequality in any place implies equalities to its right and left. Over general local rings, we construct classes of modules that show that a strict inequality can occur at almost every place in the sequence.     

New Characterizations of Dualizing Modules

Let R be a commutative Noetherian ring. In this article, we provide some new criteria for a semidualizing module to be dualizing in terms of special homological properties of module categories. The purpose of this article is twofold: first, it aims at improving Christensen's and Takahashi et al.'s characterizations of dualizing modules; secondly, while applying these criteria to the ring itself, we not only recover some results of Jenda and Xu, respectively, but also obtain a new characterization of Gorenstein rings.     

Gorenstein Injective and Gorenstein Flat Resolution of Modules Over Gorenstein Rings

Abstract

Let R be a commutative Noetherian ring. There are several characterizations of Gorenstein rings in terms of classical homological dimensions of their modules. In this paper, we use Gorenstein dimensions (Gorenstein injective and Gorenstein flat dimension) to describe Gorenstein rings. Moreover a characterization of Gorenstein injective (resp. Gorenstein flat) modules over Gorenstein rings is given in terms of their Gorenstein flat (resp. Gorenstein injective) resolutions.     

Relative Gorenstein Dimensions

In the last years (Gorenstein) homological dimensions relative to a semidualizing module C have been subject of several works as interesting extensions of (Gorenstein) homological dimensions. In this paper, we extend to the noncommutative case the concepts of G _C -projective module and dimension, weakening the condition of C being semidualizing as well. We prove that indeed they share the principal properties of the classical ones and relate this new dimension with the classical Gorenstein projective dimension of a module. The dual concepts of G _C -injective modules and dimension are also treated. Finally, we show some interesting interactions between the class of G _C -projective modules and the Bass class associated to C on one side, and the class of G\({_{C^{\vee}}}\) -injective modules (C ^∨ = Hom _R (C, E) where E is an injective cogenerator in R-Mod) and the Auslander class associated to C in the other.     

Totally reflexive modules with respect to a semidualizing bimodule

Let S and {iaR} be two associative rings, let _S C _R be a semidualizing (S,R)-bimodule. We introduce and investigate properties of the totally reflexive module with respect to _S C _R and we give a characterization of the class of the totally C _R -reflexive modules over any ring R. Moreover, we show that the totally C _R -reflexive module with finite projective dimension is exactly the finitely generated projective right R-module. We then study the relations between the class of totally reflexive modules and the Bass class with respect to a semidualizing bimodule. The paper contains several results which are new in the commutative Noetherian setting.     

THE P-COTORSION DIMENSIONS OF MODULES AND RINGS

Let R be a ring. We define a dimension, called P-cotorsion dimension, for modules and rings. The aim of this article is to investigate P-cotorsion dimensions of modules and rings and the relations between P-cotorsion dimension and other homological dimensions. This dimension has nice properties when the ring in consideration is generalized morphic.     

Characterizations of some rings with <Emphasis Type="Italic">C</Emphasis>-projective, <Emphasis Type="Italic">C</Emphasis>-(FP)-injective and <Emphasis Type="Italic">C</Emphasis>-flat modules

Let R be a commutative ring and C a semidualizing R-module. We investigate the relations between C-flat modules and C-FP-injective modules and use these modules and their character modules to characterize some rings, including artinian, noetherian and coherent rings.     

相对于余挠对的维数(英文)

本文介绍了右R-模的F-维数(C-维数)以及环R上整体F-维数(C-维数).利用同调方法,给出了平坦模维数的新刻画.另外,得到了von Neumann正则环和完全环的新刻画.     

On s-Semipermutable Maximal and Minimal Subgroups of Sylow p-Subgroups of Finite Groups

Let R be a Noetherian algebra over a field k. A formula is given for the Krull dimension of the ring R?_k k(X) in terms of the heights of simple modules with large endomorphism rings.     

On Rings Having McCoy-Like Conditions

ABSTRACT

In this paper, we study the behavior of the couniform (or dual Goldie) dimension of a module under various polynomial extensions. For a ring automorphism σ ∈ Aut(R), we use the notion of a σ-compatible module M _R to obtain results on the couniform dimension of the polynomial modules M[x], M[x ^?1], and M[x, x ^?1] over suitable skew extension rings.     

Almost finite modules

Assume that ?(R, m, k) → (S, n, l) is a local homomorphism between commutative noetherian local rings R and S. We say that an S-module M is almost finite over R if it is finitely generated over S (the R-structure on M is induced by ?). We investigate the homological behaviour of such modules, as well as various properties of the rings R and S in the presence of an almost finite module of finite flat dimension over R.     

Remarks on torsionfreeness and its applications

In this article, we shall characterize torsionfreeness of modules with respect to a semidualizing module in terms of the Serre’s condition (S_n). As its applications, we give a characterization of Cohen-Macaulay rings R such that R_𝔭 is Gorenstein for all prime ideals 𝔭 of height less than n, and we will give a partial answer of Tachikawa conjecture and Auslander-Reiten conjecture.     

Some criteria for the Cohen-Macaulay property and local cohomology

Let a be an ideal of a commutative Noetherian ring R and M be a finitely generated R-module of dimension d. We characterize Cohen-Macaulay rings in term of a special homological dimension. Lastly, we prove that if R is a complete local ring, then the Matlis dual of top local cohomology module Ha^d（M） is a Cohen-Macaulay R-module provided that the R-module M satisfies some conditions.     

On projective embeddings of generic rational surfaces

Let R be a commutative ring and let ?(σ) be a Gabriel filter of R such that R is σ-noetherian. We discuss the decomposition of the σ ¹-torsion submodule of a σ-torsionfree R-module and characterize the σ ^l-injectivity of σ-closed R-modules through the σ _m-injectivity of modules over noetherian local rings (S, m). As an application, we obtain new criteria to determine injectivity of modules over noetherian rings, of finite Krull dimension, and Krull domains.     

Gorenstein Injective and Projective Complexes with Respect to a Semidualizing Module

Let R be a commutative (possibly non-Noetherian) ring (in order to make things less technical) and C a semidualizing R-module. In this article, we introduce and investigate the notion of G _C -injective (G _C -projective) complexes. This extends Enochs and García Rozas's notion of Gorenstein injective (Gorenstein projective) complexes. We then show that a complex X is G _C -injective (G _C -projective) if and only if X ^m is a G _C -injective (G _C -projective) module for each m ∈ ?.     

Gorenstein Homological Dimension with Respect to a Semidualizing Module and a Generalization of a Theorem of Bass

Let C be a semidualizing module for a commutative ring R. In this paper, we study the resulting modules of finite G _C -projective dimension in Bass class, showing that they admit G _C -projective precover. Over local ring, we prove that dim _R (M) ≤ 𝒢? _C  ? id _R (M) for any nonzero finitely generated R-module M, which generalizes a result due to Bass.     

On the Regularity and Koszulness of Modules Over Local Rings

Koszul modules over Noetherian local rings R were introduced by Herzog and Iyengar and they possess good homological properties, for instance their Poincaré series is rational. It is an interesting problem to characterize classes of Koszul modules. Following the idea traced by Avramov, Iyengar, and Sega, we take advantage of the existence of special filtration on R for proving that large classes of R-modules over Koszul rings are Koszul modules. By using this tool we reprove and extend some results obtained by Fitzgerald.     

Complete intersection dimension

A new homological invariant is introduced for a finite module over a commutative noetherian ring: its CI-dimension. In the local case, sharp quantitative and structural data are obtained for modules of finite CI-dimension, providing the first class of modules of (possibly) infinite projective dimension with a rich structure theory of free resolutions. The first author was partly supported by NSF Grant No. DMS-9102951.