Finitely Generated Flat Modules and a Characterization of Semiperfect Rings

Abstract

For a ring S, let K ₀(FGFl(S)) and K ₀(FGPr(S)) denote the Grothendieck groups of the category of all finitely generated flat S-modules and the category of all finitely generated projective S-modules respectively. We prove that a semilocal ring Ris semiperfect if and only if the group homomorphism K ₀(FGFl(R)) → K ₀(FGFl(R/J(R))) is an epimorphism and K ₀(FGFl(R)) = K ₀(FGPr(R)).     

Gorenstein projective dimensions of complexes

We show that over a right coherent left perfect ring R, a complex C of left R-modules is Gorenstein projective if and only if C ^m is Gorenstein projective in R-Mod for all m ∈ ℤ. Basing on this we show that if R is a right coherent left perfect ring then Gpd(C) = sup{Gpd(C ^m )|m ∈ ℤ} where Gpd(−) denotes Gorenstein projective dimension.     

Cone Length for DG Modules and Global Dimension of DG Algebras

As the definition of free class of differential modules over a commutative ring in [1 Avramov , L. L. , Buchweitz , R.-O. , Iyengar , S. ( 2007 ). Class and rank of differential modules . Invent. Math. 169 : 1 – 35 .[Crossref], [Web of Science ®] , [Google Scholar]], we define DG free class for semifree DG modules over an Adams connected DG algebra A. For any DG A-modules M, we define its cone length as the least DG free classes of all semifree resolutions of M. The cone length of a DG A-module plays a similar role as projective dimension of a module over a ring does in homological ring theory. The left (resp., right) global dimension of an Adams connected DG algebra A is defined as the supremum of the set of cone lengths of all DG A-modules (resp., A ^op -modules). It is proved that the definition is a generalization of that of graded algebras. Some relations between the global dimension of H(A) and the left (resp. right) global dimension of A are discovered. When A is homologically smooth, we prove that the left (right) global dimension of A is finite and the dimension of D(A) and D ^c (A) are not bigger than the DG free class of a minimal semifree resolution X of the DG A ^e -module A.     

Two Commuting Involutions Fixing RP1(2m + 1) S RP2(2m + 1)*

Let Z₂ denote a cyclic group of 2 order and Z₂² = Z₂ ×Z₂ the direct product of groups. Suppose that (M, Φ) is a closed and smooth manifold M with a smooth Z₂² -action whose fixed point set is the disjoint union of two real projective spaces with the same dimension. In this paper, the authors give a sufficient condition on the fixed data of the action for (M, Φ) bounding equivariantly.     

Preservation of quasi-isomorphisms of complexes

We consider the preservation property of the homomorphism and tensor product functors for quasi-isomorphisms and equivalences of complexes.Let X and Y be two classes of R-modules with Ext ≥1(X,Y) = 0 for each object X ∈X and each object Y ∈Y.We show that if A,B ∈C■(R) are X-complexes and U,V ∈ C■(R) are Y-complexes,then U■V■Hom(A,U)■Hom(A,V);A■B■Hom(B,U)■Hom(A,U).As an application,we give a sufficient condition for the Hom evaluation morphism being invertible.     

Estimates of global dimension

In this note we show that for a *ⁿ-module, in particular, an almost n-tilting module, P over a ring R with A = End_R P such that P _A has finite flat dimension, the upper bound of the global dimension of A can be estimated by the global dimension of R and hence generalize the corresponding results in tilting theory and the ones in the theory of *-modules. As an application, we show that for a finitely generated projective module over a VN regular ring R, the global dimension of its endomorphism ring is not more than the global dimension of R.     

Global dimension and left derived functors of Hom

It is well known that the right global dimension of a ring R is usually computed by the right derived functors of Hom and the left projective resolutions of right R-modules. In this paper, for a left coherent and right perfect ring R, we characterize the right global dimension of R, from another point of view, using the left derived functors of Hom and the right projective resolutions of right R-modules. It is shown that rD(R)≤n (n≥2) if and only if the gl right Proj-dim MR≤n - 2 if and only if Extn-1(N, M) = 0 for all right R-modules N and M if and only if every (n - 2)th Proj-cosyzygy of a right R-module has a projective envelope with the unique mapping property. It is also proved that rD(R)≤n (n≥1) if and only if every (n-1)th Proj-cosyzygy of a right R-module has an epic projective envelope if and only if every nth Vroj-cosyzygy of a right R-module is projective. As corollaries, the right hereditary rings and the rings R with rD(R)≤2 are characterized.     

Ascent properties for pairs of modules

Given a flat local ring homomorphism \({R \rightarrow S}\) and two finitely generated R-modules M and N, we describe conditions under which the modules \({{\rm Tor}^{R}_{i}(M,N)}\) and \({{\rm Ext}^{i}_{R}(M,N)}\) have S-module structures that are compatible with their R-module structures.     

GP-projective and GI-injective modules

Let R be a ring. This paper introduces and studies GP-projective and GI-injective left R-modules. Our main goal is to investigate the “global” dimension $GPID(R) = \sup \{ id(M)|M \in _R \mathcal{M},Mis Gorenstein projective\} .$      

Gorenstein Modules and Dualizing Modules

In this article, we study the characterizations of Gorenstein injective left S-modules and finitely generated Gorenstein projective left R-modules when there is a dualizing S-R-bimodule associated with a right noetherian ring R and a left noetherian ring S.     

Gorenstein AC-projective dimension of unbounded complexes

Recently, the notion of Gorenstein AC-projective (resp., Gorenstein AC-injective) modules was introduced in [3 Bravo, D., Gillespie, J., Hovey, M. The stable module category of a general ring. http://arxiv.org/abs/1405.5768. [Google Scholar]] by which the so-called “Gorenstein AC-homological algebra” was established. Here, we define and study a notion of Gorenstein AC-projective dimension for complexes (not necessarily bounded) over associative rings, which is inspired by Veliche’s construction of defining Gorenstein projective dimension. In particular, we show that such a dimension can be closely related to the “proper” Gorenstein AC-projective resolutions of complexes induced by a complete and hereditary cotorsion pair in the category of complexes of modules. This enables us to interpret this dimension of a complex in terms of vanishing of the derived functor RHom_R(?,?). As applications, some characterizations of the Gorenstein AC-projective dimension of a module are also obtained.     

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.     

Strongly Flat Modules over Matlis Domains

Strongly flat modules were introduced by Bazzoni–Salce [3 Bazzoni , S. , Salce , L. ( 2003 ). Almost perfect domains . Colloq. Math. 95 : 285 – 301 .[Crossref] , [Google Scholar]] and used to characterize almost perfect domains. Here we wish to study strongly flat modules, more generally, over Matlis domains; these are integral domains R such that the field of quotients Q has projective dimension 1. In Section 2, criteria are proved for strong flatness. We also prove that over arbitrary domains, strongly flat submodules of projective modules are projective (Theorem 3.2), in particular, strongly flat ideals are projective (Corollary 3.4) and use these results to show that the strongly flat dimension (which makes sense over Matlis domains) coincides with the projective dimension whenever it is > 1.