Cotorsion Pairs Research Articles

On the relation between n-cotorsion pairs and (n + 1)-cluster tilting subcategories

A notion of [Formula: see text]-cotorsion pairs in an extriangulated category with enough projectives and enough injectives is defined in this paper. We show that there exists a one-to-one correspondence between [Formula: see text]-cotorsion pairs and [Formula: see text]-cluster tilting subcategories. As an application, this result generalizes the work by Huerta, Mendoza and Pérez in an abelian case. Finally, we give some examples illustrating our main result.

Ideal cotorsion theories in triangulated categories

We study ideal cotorsion pairs associated to almost exact structures in extension closed subcategories of triangulated categories. This approach allows us to extend the recent ideal approximation theory developed by Fu, Herzog et al. for exact categories in the above mentioned context. In the last part of the paper we apply the theory in order to study projective classes (in particular localization or smashing subcategories) in compactly generated triangulated categories.

N-Cotorsion pairs

Motivated by some properties satisfied by Gorenstein projective and Gorenstein injective modules over an Iwanaga-Gorenstein ring, we present the concept of left and right $n$-cotorsion pairs in an abelian category $\mathcal{C}$. Two classes $\mathcal{A}$ and $\mathcal{B}$ of objects of $\mathcal{C}$ form a left $n$-cotorsion pair $(\mathcal{A,B})$ in $\mathcal{C}$ if the orthogonality relation $\mathsf{Ext}^i_{\mathcal{C}}(\mathcal{A,B}) = 0$ is satisfied for indexes $1 \leq i \leq n$, and if every object of $\mathcal{C}$ has a resolution by objects in $\mathcal{A}$ whose syzygies have $\mathcal{B}$-resolution dimension at most $n-1$. This concept and its dual generalise the notion of complete cotorsion pairs, and has an appealing relation with left and right approximations, especially with those having the so called unique mapping property. The main purpose of this paper is to describe several properties of $n$-cotorsion pairs and to establish a relation with complete cotorsion pairs. We also give some applications in relative homological algebra, that will cover the study of approximations associated to Gorenstein projective, Gorenstein injective and Gorenstein flat modules and chain complexes, as well as $m$-cluster tilting subcategories.

Relative cohomology of complexes II: Vanishing of relative cohomology

Let R be a ring such that (GP,GP⊥) forms a cotorsion pair cogenerated by a set, where GP denotes the category of all Gorenstein projective R-modules. Recently, the first author defined for any complex X the relative cohomology functors ExtGP⁎(X,−) as H−⁎(Hom(G,−)) in which G is a special Gorenstein projective precover of X. In the present paper, we introduce the dimension of X related to Gorenstein projective precovers, and show that such a dimension of X is equal to the least integer n for which ExtGPi(X,Q)=0 for all i>n and all R-modules Q∈GP⊥. This result gives a “Gorenstein” version of the relationship between the projective dimension of complexes introduced by Avramov and Foxby and the absolutely cohomology functors ExtR⁎(−,−).

Relative Gorenstein objects in abelian categories

Abstract Let be an abelian category. For a pair of classes of objects in we define the weak and the -Gorenstein relative projective objects in We point out that such objects generalize the usual Gorenstein projective objects and other generalizations appearing in the literature as Ding-projective, Ding-injective, -Gorenstein projective, Gorenstein AC-projective and GC -projective modules and Cohen-Macaulay objects in abelian categories. We show that the principal results on Gorenstein projective modules remain true for the weak and the -Gorenstein relative objects. Furthermore, by using Auslander-Buchweitz approximation theory, a relative version of Gorenstein homological dimension is developed. Finally, we introduce the notion of -cotilting pair in the abelian category which is very strong connected with the cotorsion pairs related with relative Gorenstein objects in It is worth mentioning that the -cotilting pairs generalize the notion of cotilting objects in the sense of L. Angeleri Hügel and F. Coelho [3].

Dimension and level of triangulated categories

For the bounded derived category of an abelian category, bounds of the dimension with respect to a complete hereditary cotorsion pair are given. We also characterize levels of DG-modules and study how levels involved in a recollement of derived categories over DG-rings are related.

Integral and quasi-abelian hearts of twin cotorsion pairs on extriangulated categories

It was shown recently that the heart of a twin cotorsion pair on an extriangulated category is semi-abelian. We provide a sufficient condition for the heart to be integral and another for the heart to be quasi-abelian. This unifies and improves the corresponding results for exact and triangulated categories. Furthermore, if then we show that the Gabriel-Zisman localization of at the class of its regular morphisms is equivalent to the heart of the single twin cotorsion pair This generalizes and improves the known result for triangulated categories, thereby providing new insights in the exact setting.

Slightly (m, n)-coherent rings and (m, n)-homological dimensions

Let R be a ring and m, n two fixed positive integers. In this article, R is defined to be left slightly (m, n)-coherent if each n-generated submodule of the left R-module Rm is (m, n)-projective. It is shown that R is left slightly (m, n)-coherent if and only if the cotorsion pair is hereditary if and only if R is left (m, n)-coherent and the cotorsion pair is hereditary, where (resp., ) stands for the class of all (m, n)-projective (resp., (m, n)-injective) left R-modules, and (resp., denotes the classes of all (m, n)-flat (resp., (m, n)-cotorsion) right R-modules. As applications, (m, n)-homological dimensions of modules and rings over slightly (m, n)-coherent rings are studied.

Cograde Conditions and Cotorsion Pairs

Let R and S be rings and _R\omega_S a semidualizing bimodule. We investigate when the double functor \mathrm {Tor}^S_i(\omega, \mathrm {Ext}^i_{R}(\omega,-)) preserves epimorphisms and the double functor \mathrm {Ext}_{R}^i(\omega, \mathrm {Tor}_i^{S}(\omega,-)) preserves monomorphisms in terms of the (strong) cograde conditions of modules. Under certain cograde condition of modules, we construct two complete cotorsion pairs. In addition, we establish the relation between some relative finitistic dimensions of rings and the right and left projective dimensions of \omega .

Abelian hearts of twin cotorsion pairs

The heart of a cotorsion pair, which is a generalization of the heart of a t-structure, has been proven to be abelian on triangulated, exact, and extriangulated categories. On the other hand, the heart of a twin cotorsion pair is not always abelian (it is only semi-abelian). In this article, we study a special kind of hearts of twin cotorsion pairs induced by d-cluster tilting subcategories. We will give a sufficient-necessary condition when such hearts become abelian. We will also see that such hearts are hereditary.

Cotorsion pairs and a K-theory localization theorem

We show that a complete hereditary cotorsion pair (C,C⊥) in an exact category E, together with a subcategory Z⊆E containing C⊥, determines a Waldhausen category structure on the exact category C, in which Z is the class of acyclic objects.This allows us to prove a new version of Quillen's Localization Theorem, relating the K-theory of exact categories A⊆B to that of a cofiber. The novel idea in our approach is that, instead of looking for an exact quotient category that serves as the cofiber, we produce a Waldhausen category, constructed through a cotorsion pair. Notably, we do not require A to be a Serre subcategory, which produces new examples.Due to the algebraic nature of our Waldhausen categories, we are able to recover a version of Quillen's Resolution Theorem, now in a more homotopical setting that allows for weak equivalences.

Relatively divisible and relatively flat objects in exact categories

We introduce and study relatively divisible and relatively flat objects in exact categories in the sense of Quillen. For every relative cotorsion pair [Formula: see text] in an exact category [Formula: see text], [Formula: see text] coincides with the class of relatively flat objects of [Formula: see text] for some relative projectively generated exact structure, while [Formula: see text] coincides with the class of relatively divisible objects of [Formula: see text] for some relative injectively cogenerated exact structure. We exhibit Galois connections between relative cotorsion pairs in exact categories, relative projectively generated exact structures and relative injectively cogenerated exact structures in additive categories. We establish closure properties and characterizations in terms of the approximation theory.

Singular compactness and definability for $$\Sigma $$-cotorsion and Gorenstein modules

We introduce a general version of singular compactness theorem which makes it possible to show that being a $\Sigma$-cotorsion module is a property of the complete theory of the module. As an application of the powerful tools developed along the way, we give a new description of Gorenstein flat modules which implies that, regardless of the ring, the class of all Gorenstein flat modules forms the left-hand class of a perfect cotorsion pair. We also prove the dual result for Gorenstein injective modules.

The balancedness of N -complexes based on cotorsion pairs

The balancedness of N -complexes based on cotorsion pairs

The derived category with respect to $\boldsymbol{\mathcal{G}(\mathcal{X})}$

Let $\\mathcal{A}$ be an abelian category with enough projective objects and $(\\mathcal{X},~\\mathcal{Y})$ be a complete and hereditary cotorsion pair in $\\mathcal{A}$. We introduce the Gorenstein $\\mathcal{X}$-derived category, denoted by $\\mathbf{D}_{\\mathcal{G}(\\mathcal{X})}{(\\mathcal{A})}$, and investigate $\\mathbf{D}_{\\mathcal{G}(\\mathcal{X})}{(\\mathcal{A})}$ from different perspectives. When $(\\mathcal{X},~\\mathcal{Y})$ is served as a special cotorsion pair, we obtain some known derived categories, such as Gorenstein derived categories, Gorenstein flat derived categories and so on. The bounded Gorenstein $\\mathcal{X}$-derived category $\\mathbf{D}^{b}{_{\\mathcal{G}(\\mathcal{X})}{(\\mathcal{A})}}$ and bounded derived category $\\mathbf{D}^{b}{(\\mathcal{A})}$ are described via the homology category $\\mathbf{K}^{-,gxb}{(\\mathcal{G}(\\mathcal{X}))}$ related to $\\mathcal{G}(\\mathcal{X})$ and some triangle equivalences are given.

Gorenstein category with respect to twin cotorsion pair

Gorenstein category with respect to twin cotorsion pair

Cotorsion pairs and adjoint functors in the homotopy category of N-complexes

In this paper, we first construct some complete cotorsion pairs on the category [Formula: see text] of unbounded [Formula: see text]-complexes of Grothendieck category [Formula: see text], from two given cotorsion pairs in [Formula: see text]. Next, as an application, we focus on particular homotopy categories and the existence of adjoint functors between them. These are an [Formula: see text]-complex version of the results that were shown by Neeman in the category of ordinary complexes.

English

We introduce the right (left) Gorenstein subcategory relative to an additive subcategory $\C$ of an abelian category $\A$, and prove that the right Gorenstein subcategory $r\mathcal{G}(\mathscr{C})$ is closed under extensions, kernels of epimorphisms, direct summands and finite direct sums. When $\C$ is self-orthogonal, we give a characterization for objects in $r\mathcal{G}(\mathscr{C})$, and prove that any object in $\A$ with finite $r\mathcal{G}(\C)$-projective dimension is isomorphic to a kernel (resp. a cokernel) of a morphism from an object in $\A$ with finite $\C$-projective dimension to an object in $r\mathcal{G}(\C)$. As an application, we obtain a weak Auslander-Buchweitz context related to the kernel of a hereditary cotorsion pair in $\A$ having enough injectives.

Cotorsion pairs and approximation classes over formal triangular matrix rings

Let T=(A0UB) be a formal triangular matrix ring, where A and B are rings and U is a (B,A)-bimodule. Let C1 and C2 be two classes of left A-modules, D1 and D2 be two classes of left B-modules, we prove that: (1) If ToriA(U,C1)=0 for any i≥1, (C1,C2) and (D1,D2) are (resp. hereditary complete) cotorsion pairs, then (PD1C1,AD2C2) is a (resp. hereditary complete) cotorsion pair in T-Mod. (2) If ExtBi(U,D2)=0 for any i≥1, (C1,C2) and (D1,D2) are (resp. hereditary complete) cotorsion pairs, then (AD1C1,ID2C2) is a (resp. hereditary complete) cotorsion pair in T-Mod. In addition, we characterize some special preenveloping classes and precovering classes in T-Mod.

Abelian model structures on categories of quiver representations

Let [Formula: see text] be an abelian model category (in the sense of Hovey). For a large class of quivers, we describe associated abelian model structures on categories of quiver representations with values in [Formula: see text]. This is based on recent work of Holm and Jørgensen on cotorsion pairs in categories of quiver representations. An application on Ding projective and Ding injective representations of quivers over Ding–Chen rings is given.