Finitistic dimensions of semiprimary rings

LetR be a semiprimary ring. We show that if the left generalized projective dimension (defined below) of _R (R/J ²) is finite, then the injectively defined left finitistic dimension ofR is finite.     

Gorenstein injective envelopes and covers over two sided noetherian rings

We prove that the class of Gorenstein injective modules is both enveloping and covering over a two sided noetherian ring such that the character modules of Gorenstein injective modules are Gorenstein flat. In the second part of the paper we consider the connection between the Gorenstein injective modules and the strongly cotorsion modules. We prove that when the ring R is commutative noetherian of finite Krull dimension, the class of Gorenstein injective modules coincides with that of strongly cotorsion modules if and only if the ring R is in fact Gorenstein.     

On Copure Projective Modules and Copure Projective Dimensions

Let R be any ring. A right R-module M is called n-copure projective if Ext¹(M, N) = 0 for any right R-module N with fd(N) ≤ n, and M is said to be strongly copure projective if Ext ⁱ (M, F) = 0 for all flat right R-modules F and all i ≥ 1. In this article, firstly, we present some general properties of n-copure projective modules and strongly copure projective modules. Then we define and investigate copure projective dimensions of modules and rings. Finally, more properties and applications of n-copure projective modules, strongly copure projective modules and copure projective dimensions are given over coherent rings with finite self-FP-injective dimension.     

Gorenstein Flat Covers and Gorenstein Cotorsion Modules Over Integral Group Rings

Every module over an Iwanaga–Gorenstein ring has a Gorenstein flat cover [13] (however, only a few nontrivial examples are known). Integral group rings over polycyclic-by-finite groups are Iwanaga–Gorenstein [10] and so their modules have such covers. In particular, modules over integral group rings of finite groups have these covers. In this article we initiate a study of these covers over these group rings. To do so we study the so-called Gorenstein cotorsion modules, i.e. the modules that split under Gorenstein flat modules. When the ring is ℤ, these are just the usual cotorsion modules. Harrison [16] gave a complete characterization of torsion free cotorsion ℤ-modules. We show that with appropriate modifications Harrison's results carry over to integral group rings ℤG when G is finite. So we classify the Gorenstein cotorsion modules which are also Gorenstein flat over these ℤG. Using these results we classify modules that can be the kernels of Gorenstein flat covers of integral group rings of finite groups. In so doing we necessarily give examples of such covers. We use the tools we develop to associate an integer invariant n with every finite group G and prime p. We show 1≤n≤|G : P| where P is a Sylow p-subgroup of G and gives some indication of the significance of this invariant. We also use the results of the paper to describe the co-Galois groups associated to the Gorenstein flat cover of a ℤG-module. Presented by A. Verschoren Mathematics Subject Classifications (2000) 20C05, 16E65.     

Finitistic dimension and Igusa–Todorov algebras

The notion of Igusa–Todorov algebras is introduced in connection with the (little) finitistic dimension conjecture, and the conjecture is proved for those algebras. Such algebras contain many known classes of algebras over which the finitistic dimension conjecture holds, e.g., algebras with the representation dimension at most 3, algebras with radical cube zero, monomial algebras and left serial algebras, etc. It is an open question whether all artin algebras are Igusa–Todorov. We provide some methods to construct many new classes of (2-)Igusa–Todorov algebras and thus obtain many algebras such that the finitistic dimension conjecture holds. In particular, we show that the class of 2-Igusa–Todorov algebras is closed under taking endomorphism algebras of projective modules. Hence, if all quasi-hereditary algebras are 2-Igusa–Todorov, then all artin algebras are 2-Igusa–Todorov by [V. Dlab, C.M. Ringel, Every semiprimary ring is the endomorphism ring of a projective module over a quasihereditary ring, Proc. Amer. Math. Soc. 107 (1) (1989) 1–5] and have finite finitistic dimension.     

Answer to a conjecture of J.M. Cohen

The origin of Gelfand rings comes from [9] where the Jacobson topology and the weak topology are compared. The equivalence of these topologies defines a regular Banach algebra. One of the interests of these rings resides in the fact that we have an equivalence of categories between vector bundles over a compact manifold and finitely generated projective modules over C(M), the ring of continuous real functions on M [17].These rings have been studied by R. Bkouche (soft rings [3]) C.J. Mulvey (Gelfand rings [15]) and S. Teleman (harmonic rings [19]).Firstly we study these rings geometrically (by sheaves of modules (Theorem 2.5)) and then introduce the ?ech covering dimension of their maximal spectrums. This allows us to study the stable rank of such a ring A (Theorem 6.1), the nilpotence of the nilideal of K₀(A) - The Grothendieck group of the category of finitely generated projective A-modules - (Theorem 9.3), and an upper limit on the maximal number of generators of a finitely generated A-module as a function of the afore-mentioned dimension (Theorem 4.4).Moreover theorems of stability are established for the group K₀(A), depending on the stable rank (Theorems 8.1 and 8.2). They can be compared to those for vector bundles over a finite dimensional paracompact space [18].Thus there is an analogy between finitely generated projective modules over Gelfand rings and ?ech dimension, and finitely generated projective modules over noetherian rings and Krull dimension.     

Face rings of simplicial complexes with singularities

The face ring of a simplicial complex modulo m generic linear forms is shown to have finite local cohomology if and only if the link of every face of dimension m or more is nonsingular, i.e., has the homology of a wedge of spheres of the expected dimension. This is derived from an enumerative result for local cohomology of face rings modulo generic linear forms, as compared with local cohomology of the face ring itself. The enumerative result is generalized to squarefree modules. A concept of Cohen–Macaulay in codimension c is defined and characterized for arbitrary finitely generated modules and coherent sheaves. For the face ring of an r-dimensional complex Δ, it is equivalent to nonsingularity of Δ in dimension r − c; for a coherent sheaf on projective space, this condition is shown to be equivalent to the same condition on any single generic hyperplane section. The characterization of nonsingularity in dimension m via finite local cohomology thus generalizes from face rings to arbitrary graded modules.     

FINITELY GENERATED MODULES AND POWER STABLY FREE DIMENSION

In this paper, we define the power stably free dimension for rings. Using its relations with other dimensions, we get a classification of rings. Moreover, we give the equivalent characterizations of a ring with power stably free dimension 0, 1 respectively.

For a commutative ring R in which each f. g. module has a finite power stably free dimension, we show that R[x ₁, …, x_n ] is connected and all f. g. projective modules over R[x ₁, …, x_n ] are power free.     

COTORSION MODULES AND PROJECTIVE DIMENSION ONE

If T is a perfect torsion theory for a category of modules over a commutative ring R, a module C is called T—cotorsion provided Hom_R(Q_T,C) = 0 = Ext_R (Q_T,C) where Q_T denotes the T-injective hull of R. Motivated by the now classical results of D. K. Harrison for abelian groups and of E. Matlis for modules over a domain, the theory of T—cotorsion modules is extended. For example, a category equivalence is obtained between the category of T—compact T-cotorsion modules and the category of T-torsion T-reduced modules. The class of T-divisible modules (homomorphic images of T-injective modules) is shown to be closed under formation of extensions if and only if pd_RQ_T ≤ 1, in the case that Q_T is T—cocritical.     

Self-Orthogonal Modules of Finite Projective Dimension

Let R be a ring and _R ω a self-orthogonal module. We introduce the notion of the right orthogonal dimension (relative to _R ω) of modules. We give a criterion for computing this relative right orthogonal dimension of modules. For a left coherent and semilocal ring R and a finitely presented self-orthogonal module _R ω, we show that the projective dimension of _R ω and the right orthogonal dimension (relative to _R ω) of R/J are identical, where J is the Jacobson radical of R. As a consequence, we get that _R ω has finite projective dimension if and only if every left (finitely presented) R-module has finite right orthogonal dimension (relative to _R ω). If ω is a tilting module, we then prove that a left R-module has finite right orthogonal dimension (relative to _R ω) if and only if it has a special ω^⊥-preenvelope.     

Another type of dimensions of modules and rings

Abstract

We say that a class Q of left R-modules is a monic class if a nonzero submodule of a module in Q is also a module in Q. For a monic class Q, we define a Q-dimension of modules that measures how far modules are from the modules in Q. For a monic class Q of indecomposable modules we characterize rings whose modules have Q-dimension. We prove that for an artinian principal ideal ring the Q-dimension coincides with the uniserial dimension. We also characterize when every module has Q-dimension.     

On the Left Perpendicular Category of the Modules of Finite Projective Dimension

In this article, we characterize several properties of commutative noetherian local rings in terms of the left perpendicular category of the category of finitely generated modules of finite projective dimension. As an application, we prove that a local ring is regular if (and only if) there exists a strong test module for projectivity having finite projective dimension. We also obtain corresponding results with respect to a semidualizing module.     

Finitistic Extension Degree

We introduce the finitistic extension degree of a ring and investigate rings for which it is finite. The Auslander–Reiten Conjecture is proved for rings of finite finitistic extension degree and these rings are also shown to have finite finitistic dimension. We apply these results to better understand a generalized version of the Auslander–Reiten Condition for Gorenstein rings. We also record how the finitistic extension degree behaves with respect to many change of ring procedures that arise frequently in the commutative setting.     

Homological dimensions and semidualizing complexes

For finite modules over a local ring and complexes with finitely generated homology, we consider several homological invariants sharing some basic properties with projective dimension. In the second section, we introduce the notion of a semidualizing complex, which is a generalization of both a dualizing complex and a suitable module. Our goal is to establish some common properties of such complexes and the homological dimension with respect to them. Basic properties are investigated in Sec. 2.1. In Sec. 2.2, we study the structure of the set of semidualizing complexes over a local ring, which is closely related to the conjecture of Avramov-Foxby on the transitivity of the G-dimension. In particular, we prove that, for a pair of semidualizing complexes X ₁ and X ₂ such that G _X2, we have X ₂ ≃ X ₁ ⊗ _R ^L RHom _R (X ₁, X ₂). Specializing to the case of semidualizing modules over Artinian rings, we obtain a number of quantitative results for the rings possessing a configuration of semidualizing modules of special form. For the rings with m ³=0, this condition reduces to the existence of a nontrivial semidualizing module, and we prove a number of structural results in this case. In the third section, we consider the class of modules that contains the modules of finite CI-dimension and enjoys some nice additional properties, in particular, good behavior in short exact sequences. In the fourth section, we introduce a new homological invariant, CM-dimension, which provides a characterization for Cohen-Macaulay rings in precisely the same way as projective dimension does for regular rings, CI-dimension for locally complete intersections, and G-dimension for Gorenstein rings. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 30, Algebra, 2005.     

On semilocal modules and rings

It is well-known that a ring Ris semiperfect if and only if _RR (orR_R ) is a supplemented module. Considering weak supplementsinstead of supplements we show that weakly supplemented modules Mare semilocal (i.e.M/Rad(M) is semisimple) and that R is a semilocal ring if and only if _RR (orR_R ) is weakly supplemented. In this context the notion of finite hollow dimension (or finite dual Goldie dimension) of modules is of interest and yields a natural interpretation of the Camps-Dicks characterization of semilocal rings. Finitely generated modules are weakly supplemented if and only if they have finite hollow dimension (or are semilocal).     

Gorenstein projective and flat complexes over noetherian rings

We give sufficient conditions on a class of R‐modules $\mathcal {C}We give sufficient conditions on a class of R‐modules $\mathcal {C}$ in order for the class of complexes of $\mathcal {C}$‐modules, $dw \mathcal {C}$, to be covering in the category of complexes of R‐modules. More precisely, we prove that if $\mathcal {C}$ is precovering in R ? Mod and if $\mathcal {C}$ is closed under direct limits, direct products, and extensions, then the class $dw \mathcal {C}$ is covering in Ch(R). Our first application concerns the class of Gorenstein flat modules. We show that when the ring R is two sided noetherian, a complex C is Gorenstein flat if and only if each module C_n is Gorenstein flat. If moreover every direct product of Gorenstein flat modules is a Gorenstein flat module, then the class of Gorenstein flat complexes is covering. We consider Gorenstein projective complexes as well. We prove that if R is a commutative noetherian ring of finite Krull dimension, then the class of Gorenstein projective complexes coincides with that of complexes of Gorenstein projective modules. We also show that if R is commutative noetherian with a dualizing complex then every right bounded complex has a Gorenstein projective precover.     

Rings Over Which the Class of Gorenstein Flat Modules is Closed Under Extensions

A ring R is called left “GF-closed”, if the class of all Gorenstein flat left R-modules is closed under extensions. The class of left GF-closed rings includes strictly the one of right coherent rings and the one of rings of finite weak dimension.

In this article, we investigate the Gorenstein flat dimension over left GF-closed rings. Namely, we generalize the fact that the class of all Gorenstein flat left modules is projectively resolving over right coherent rings to left GF-closed rings. Also, we generalize the characterization of Gorenstein flat left modules (then of Gorenstein flat dimension of left modules) over right coherent rings to left GF-closed rings. Finally, using direct products of rings, we show how to construct a left GF-closed ring that is neither right coherent nor of finite weak dimension.     

$\mathcal{F}$-Perfect Rings and Modules

Let $R$ be a ring, and let $(\mathcal{F}, C)$ be a cotorsion theory. In this article, the notion of $\mathcal{F}$-perfect rings is introduced as a nontrial generalization of perfect rings and A-perfect rings. A ring $R$ is said to be right $\mathcal{F}$-perfect if $F$ is projective relative to $R$ for any $F ∈ \mathcal{F}$. We give some characterizations of $\mathcal{F}$-perfect rings. For example, we show that a ring $R$ is right $\mathcal{F}$-perfect if and only if $\mathcal{F}$-covers of finitely generated modules are projective. Moreover, we define $\mathcal{F}$-perfect modules and investigate some properties of them.     

Gorenstein Projective Precovers

We prove that the class of Gorenstein projective modules is special precovering over any left GF-closed ring such that every Gorenstein projective module is Gorenstein flat and every Gorenstein flat module has finite Gorenstein projective dimension. This class of rings includes (strictly) Gorenstein rings, commutative noetherian rings of finite Krull dimension, as well as right coherent and left n-perfect rings. In Sect. 4 we give examples of left GF-closed rings that have the desired properties (every Gorenstein projective module is Gorenstein flat and every Gorenstein flat has finite Gorenstein projective dimension) and that are not right coherent.