Gorenstein Cotorsion Research Articles

Gorenstein orthogonal classes and Gorenstein weak tilting modules

Let R be a Gorenstein ring. We first study Gorenstein orthogonal classes and Gorenstein cotorsion pairs of R-modules. Then, we introduce the concept of Gorenstein weak tilting R-modules, which is a “Gorenstein analogue” of weak tilting modules. It is proven that a left R-module W is Gorenstein weak n-tilting if and only if its character module W + is Gorenstein n-cotilting if and only if its Gorenstein flat dimension is at most n and its left Gorenstein Tor-orthogonal class consists precisely of all right R-modules with a G-exact right Prod-resolution of finite length. Finally, we explore the connections between Gorenstein weak tilting modules and Gorenstein tilting modules.

Applications of cotorsion triples

We study homotopy categories of model categories arising from a cotorsion triple, and the equivalences between corresponding stable categories. We characterize homological dimensions with respect to a cotorsion triple. Then, we lift cotorsion triple to complexes, and get the equivalence of homotopy categories of complexes via Quillen equivalence of model categories. Finally, we specify to Gorenstein cotorsion triple . By Quillen equivalence, it is shown that for a left-Gorenstein ring, there is an equivalence , which restricts to an equivalence (compare to [J. Algebra, 2010, 324:2718-2731]); a new proof for Bennis and Mahdou’s equality of Gorenstein global dimension is also given.

Weak Gorenstein Cotorsion Modules

Let R be a ring and let 𝒮𝒢ℱ be the class of strongly Gorenstein at right R-modules. We call a right R-module M a weak Gorenstein cotorsion module if M is in the class 𝒮𝒢ℱ⊥. Properties of weak Gorenstein cotorsion modules are investigated. It is shown that weak Gorenstein cotorsion R-modules over coherent rings are indeed weaker than Gorenstein cotorsion R-modules. Weak Gorenstein cotorsion dimension for modules and rings are also studied.

Relative singularity categories with respect to gorenstein flat modules

Let R be a right coherent ring and D b (R-Mod) the bounded derived category of left R-modules. Denote by \({D^b}{\left( {R - Mod} \right)_{\widehat {\left[ {GF,C} \right]}}}\) the subcategory of D b (R-Mod) consisting of all complexes with both finite Gorenstein flat dimension and cotorsion dimension and K b (F ∩ C) the bounded homotopy category of flat cotorsion left R-modules. We prove that the quotient triangulated category \({D^b}{\left( {R - Mod} \right)_{\widehat {\left[ {GF,C} \right]}}}\)/K b (F ∩ C) is triangle-equivalent to the stable category GF ∩ C of the Frobenius category of all Gorenstein flat and cotorsion left R-modules.

The flat stable module category of a coherent ring

Let R by a right coherent ring and R-Mod denote the category of left R-modules. We show that there is an abelian model structure on R-Mod whose cofibrant objects are precisely the Gorenstein flat modules. Employing a new method for constructing model structures, the key step is to show that a module is flat and cotorsion if and only if it is Gorenstein flat and Gorenstein cotorsion.

Notes on Gorenstein cotorsion modules

Let R be a ring. An R-module N is called Gorenstein cotorsion module if ExtR1 (F, N) = 0 for any Gorenstein flat R-module F. We investigate some properties of Gorenstein cotorsion envelopes and introduce Gorenstein cotorsion dimension. Some characterizations of Gorenstein cotorsion dimension of modules and rings are given.

ON GORENSTEIN COTORSION DIMENSION OVER GF-CLOSED RINGS

In this article, we introduce and study the Gorenstein cotorsion dimension of modules and rings. It is shown that this dimension has nice properties when the ring in question is left GF-closed. The relations between the Gorenstein cotorsion dimension and other homological dimensions are discussed. Finally, we give some new characterizations of weak Gorenstein global dimension of coherent rings in terms of Gorenstein cotorsion modules.

Homological dimensions of complexes related to cotorsion pairs

Let (A ; B ) be a cotorsion pair in R-Mod. We dene and study notions of A dimension and B dimension of unbounded complexes, which is given by means of dg-projective resolution and dg-injective resolution, respectively. As an application, we extend the Gorenstein at dimension of complexes, which was dened by Iacob. Gorenstein cotorsion, FP-projective, FP-injective, Ding projective, and Ding injective dimension are also extended from modules to complexes. Moreover, we characterize Noetherian rings, von Neumann regular rings, and QF rings by the FP-projective, FP-injective, and Ding projective (injective) dimension of complexes, respectively.

Gorenstein Flat and Cotorsion Dimensions of Unbounded Complexes

In this article, we define and study the Gorenstein flat dimension and Gorenstein cotorsion dimension for unbounded complexes over GF-closed rings by constructions of resolutions of unbounded complexes. The behavior of the dimensions under change of rings is investigated.

GORENSTEIN COTORSION AND FLAT COMPLEXES

A complex C is called Gorenstein cotorsion if Ext 1(G, C) = 0 for any Gorenstein flat complex G. It is shown that a complex C of left R-modules is Gorenstein cotorsion if and only if Cn is Gorenstein cotorsion in R-Mod for all n ∈ ℤ and [Formula: see text] is exact for any Gorenstein flat complex G; and [Formula: see text] is a hereditary cotorsion theory over a right coherent ring R, where [Formula: see text] and [Formula: see text] denote the classes of all Gorenstein flat and Gorenstein cotorsion complexes respectively. Also Gorenstein cotorsion envelopes and Gorenstein flat covers of complexes are considered.

Complexes of Gorenstein Flat Modules and Gorenstein Cotorsion Modules

A complex (C, δ) is called strongly Gorenstein flat if C is exact and Ker δ n is Gorenstein flat in R-Mod for all n ∈ ℤ. Let 𝒮𝒢 stand for the class of strongly Gorenstein flat complexes. We show that a complex C of left R-modules over a right coherent ring R is in the right orthogonal class of 𝒮𝒢 if and only if C n is Gorenstein cotorsion in R-Mod for all n ∈ ℤ and Hom.(G, C) is exact for any strongly Gorenstein flat complex G. Furthermore, a bounded below complex C over a right coherent ring R is in the right orthogonal class of 𝒮𝒢 if and only if C n is Gorenstein cotorsion in R-Mod for all n ∈ ℤ. Finally, strongly Gorenstein flat covers and 𝒮𝒢⊥-envelopes of complexes are considered. For a right coherent ring R, we show that every bounded below complex has a 𝒮𝒢⊥-envelope.

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.