Cotorsion Pairs Research Articles

New model structures and projective (injective) cotorsion pairs

Let [Formula: see text] be either the category of R-modules or the category of chain complexes of R-modules and [Formula: see text] a cofibrantly generated hereditary abelian model structure on [Formula: see text]. First, we get a new cofibrantly generated model structure on [Formula: see text] related to [Formula: see text] for any positive integer n, and hence, one can get new algebraic triangulated categories. Second, it is shown that any n-strongly Gorenstein projective module gives rise to a projective cotorsion pair cogenerated by a set. Finally, let M be an R-module with finite flat dimension and n a positive integer, if [Formula: see text] is an exact sequence of [Formula: see text]-modules with every [Formula: see text] Gorenstein injective, then M is injective.

Auslander's defects over extriangulated categories: An application for the general heart construction

The notion of extriangulated category was introduced by Nakaoka and Palu giving a simultaneous generalization of exact categories and triangulated categories. Our first aim is to provide an extension to extriangulated categories of Auslander's formula: for some extriangulated category $\mathcal{C}$, there exists a localization sequence $\operatorname{def}\mathcal{C} \to \mod\mathcal{C} \to \operatorname{lex}\mathcal{C}$, where $\operatorname{lex}\mathcal{C}$ denotes the full subcategory of finitely presented left exact functors and $\operatorname{def}\mathcal{C}$ the full subcategory of Auslander's defects. Moreover we provide a connection between the above localization sequence and the Gabriel–Quillen embedding theorem. As an application, we show that the general heart construction of a cotorsion pair $(\mathcal{U}, \mathcal{V})$ in a triangulated category, which was provided by Abe and Nakaoka, is the same as the construction of a localization sequence $\operatorname{def}\mathcal{U} \to \mod\mathcal{U} \to \operatorname{lex}\mathcal{U}$.

Model structures, recollements and duality pairs

In this paper, we construct some model structures corresponding Gorenstein [Formula: see text]-modules and relative Gorenstein flat modules associated to duality pairs, Frobenius pairs and cotorsion pairs. By investigating homological properties of Gorenstein [Formula: see text]-modules and some known complete hereditary cotorsion pairs, we describe several types of complexes and obtain some characterizations of Iwanaga–Gorenstein rings. Based on some facts given in this paper, we find new duality pairs and show that [Formula: see text] is covering as well as enveloping and [Formula: see text] is preenveloping under certain conditions, where [Formula: see text] denotes the class of Gorenstein [Formula: see text]-injective modules and [Formula: see text] denotes the class of Gorenstein [Formula: see text]-flat modules. We give some recollements via projective cotorsion pair [Formula: see text] cogenerated by a set, where [Formula: see text] denotes the class of Gorenstein [Formula: see text]-projective modules. Also, many recollements are immediately displayed through setting specific complete duality pairs.

Projective Covers of Flat Contramodules

Abstract We show that a direct limit of projective contramodules (over a right linear topological ring) is projective if it has a projective cover. A similar result is obtained for $\infty $-strictly flat contramodules of projective dimension not exceeding $1$, using an argument based on the notion of the topological Jacobson radical. Covers and precovers of direct limits of more general classes of objects, both in abelian categories with exact and with nonexact direct limits, are also discussed, with an eye towards the Enochs conjecture about covers and direct limits, using locally split (mono)morphisms as the main technique. In particular, we offer a simple elementary proof of the Enochs conjecture for the left class of an $n$-tilting cotorsion pair in an abelian category with exact direct limits.

Recollements arising from cotorsion pairs on extriangulated categories

This paper is devoted to constructing some recollements of additive categories associated to concentric twin cotorsion pairs on an extriangulated category. As an application, this result generalizes the work by W. J. Chen, Z. K. Liu, and X. Y. Yang in a triangulated case [J. Algebra Appl., 2018, 17(5): 1–15]. Moreover, it highlights new phenomena when it applied to an exact category. Finally, we give some applications of our main results. In particular, we obtain Krause’s recollement whose proofs are both elementary and very general.

Dimension of complexes related to special Gorenstein projective precovers

Let R be a ring and X a complex of R-modules. We investigate the relationship between the dimensions and of X, where is the dimension related to special Gorenstein projective precovers, and is Gorenstein projective dimension of X. It is shown that if R is a ring such that forms a cotorsion pair cogenerated by a set, then for any bounded complex X, there is an equality We also establish the relationship between the dimensions and where is an exact sequence of complexes.

Infinitely Generated Gorenstein Tilting Modules

The theory of finitely generated relative (co)tilting modules has been established in the 1980s by Auslander and Solberg, and infinitely generated relative tilting modules have recently been studied by many authors in the context of Gorenstein homological algebra. In this work, we build on the theory of infinitely generated Gorenstein tilting modules by developing “Gorenstein tilting approximations” and employing these approximations to study Gorenstein tilting classes and their associated relative cotorsion pairs. As applications of our results, we discuss the problem of existence of complements to partial Gorenstein tilting modules as well as some connections between Gorenstein tilting modules and finitistic dimension conjectures.

Applications of hyperhomology to adjoint functors

Hyperhomology is applied to give explicit constructions of left or right adjoint functors of some inclusions between unbounded homotopy categories of additive categories arising from cotorsion pairs in abelian categories. These constructions are independent of Brown representability and do not involve cardinality in set theory. Applications are then given to injective cotorsion pairs in abelian categories and some typical cotorsion pairs in module categories of rings.

Semibricks in extriangulated categories

Let be a semibrick in an extriangulated category Let be the filtration subcategory generated by We give a one-to-one correspondence between simple semibricks and length wide subcategories in This generalizes a bijection given by Ringel in module categories, which has been generalized by Enomoto to exact categories. Moreover, we also give a one-to-one correspondence between cotorsion pairs in and certain subsets of Applying to the simple minded systems of a triangulated category, we recover a result given by Dugas.

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.

Exact categories and infinite tilting

It is proved that any tilting adjunction is completely described by an exact category with a coherence property and the closure condition that exact sequences are acyclic. The coherence property signifies that two associated categories where the tilting adjunction takes place are abelian. The adjoint pair is then obtained in a unique fashion. Applications to tilting between module categories, n-tilting modules, tilting from cotorsion pairs, and Wakamatsu tilting modules, are given. Criteria for triangle equivalences induced by the derived functors between unbounded derived categories are established.

Covering classes and 1-tilting cotorsion pairs over commutative rings

Abstract We are interested in characterising the commutative rings for which a 1-tilting cotorsion pair ( 𝒜 , 𝒯 ) {(\mathcal{A},\mathcal{T})} provides for covers, that is when the class 𝒜 {\mathcal{A}} is a covering class. We use Hrbek’s bijective correspondence between the 1-tilting cotorsion pairs over a commutative ring R and the faithful finitely generated Gabriel topologies on R. Moreover, we use results of Bazzoni–Positselski, in particular a generalisation of Matlis equivalence and their characterisation of covering classes for 1-tilting cotorsion pairs arising from flat injective ring epimorphisms. Explicitly, if 𝒢 {\mathcal{G}} is the Gabriel topology associated to the 1-tilting cotorsion pair ( 𝒜 , 𝒯 ) {(\mathcal{A},\mathcal{T})} , and R 𝒢 {R_{\mathcal{G}}} is the ring of quotients with respect to 𝒢 {\mathcal{G}} , we show that if 𝒜 {\mathcal{A}} is covering, then 𝒢 {\mathcal{G}} is a perfect localisation (in Stenström’s sense [B. Stenström, Rings of Quotients, Springer, New York, 1975]) and the localisation R 𝒢 {R_{\mathcal{G}}} has projective dimension at most one as an R-module. Moreover, we show that 𝒜 {\mathcal{A}} is covering if and only if both the localisation R 𝒢 {R_{\mathcal{G}}} and the quotient rings R / J {R/J} are perfect rings for every J ∈ 𝒢 {J\in\mathcal{G}} . Rings satisfying the latter two conditions are called 𝒢 {\mathcal{G}} -almost perfect.

Ding injective modules

We prove that, over any ring R, the class of Ding injective modules, is the right half of a hereditary cotorsion pair, We also show that, over a coherent ring R, a module M is Ding injective if and only if it is Gorenstein injective and in the class

Models for functor categories over self-injective quivers

Let [Formula: see text] be a ring, [Formula: see text] a small [Formula: see text]-preadditive category, and let [Formula: see text] be the category of [Formula: see text]-linear functors from [Formula: see text] to the left [Formula: see text]-module category [Formula: see text]. Given a cotorsion pair [Formula: see text] in [Formula: see text], we construct four cotorsion pairs [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] in [Formula: see text] and investigate when these cotorsion pairs are hereditary and complete. Moreover, under few assumptions, we show that there exist a recollement of homotopy categories induced by these cotorsion pairs. Finally, some applications are given in the category of [Formula: see text]-periodic chain complexes of [Formula: see text]-modules.

Recollements associated to cotorsion pairs over upper triangular matrix rings

Let $A$, $B$ be two rings and $T=\left(\begin{smallmatrix} A & M 0 & B \end{smallmatrix}\right)$ with $M$ an $A$-$B$-bimodule. Given two complete hereditary cotorsion pairs $(\mathcal{A}_{A},\mathcal{B}_{A})$ and $(\mathcal{C}_{B},\mathcal{D}_{B})$ in $A$-Mod and $B$-Mod respectively. We define two cotorsion pairs $(\Phi(\mathcal{A}_{A},\mathcal{C}_{B}), \mathrm{Rep}(\mathcal{B}_{A},\mathcal{D}_{B}))$ and $(\mathrm{Rep}(\mathcal{A}_{A},\mathcal{C}_{B}), \Psi(\mathcal{B}_{A},\mathcal{D}_{B}))$ in $T$-Mod and show that both of these cotorsion pairs are complete and hereditary. Given two cofibrantly generated model structures $\mathcal{M}_{A}$ and $\mathcal{M}_{B}$ on $A$-Mod and $B$-Mod respectively. Using the result above, we investigate when there exist a cofibrantly generated model structure $\mathcal{M}_{T}$ on $T$-Mod and a recollement of $\mathrm{Ho}(\mathcal{M}_{T})$ relative to $\mathrm{Ho}(\mathcal{M}_{A})$ and $\mathrm{Ho}(\mathcal{M}_{B})$. Finally, some applications are given in Gorenstein homological algebra.

On a new cotorsion pair

In cotorsion theories, the cotorsion pairs \mathcal {SF,MC} of strongly flat and Matlis-cotorsion modules, and (\mathcal {F,EC}) of flat and Enochs-cotorsion modules play important roles.We introduce a new cotorsion pair that in general lies properly between these two (in the partial order generally accepted for cotorsion pairs), and discuss its properties over commutative rings. In particular, we characterize the commutative rings over which this is a perfect cotorsion pair. Our results may shed more light on the relation between the two old cotorsion pairs.

Test sets for factorization properties of modules

Baer's Criterion of injectivity implies that injectivity of a module is a factorization property with respect to\ a single monomorphism. Using the notion of a cotorsion pair, we study generalizations and dualizations of factorization properties in dependence on the algebraic structure of the underlying ring R and on additional set-theoretic hypotheses. For R commutative noetherian of Krull dimension 0 < d < \infty , we show that the assertion 'projectivity is a factorization property with respect to a single epimorphism' is independent of ZFC + GCH. We also show that if R is any ring and there exists a strongly compact cardinal \kappa > |R| , then the category of all projective modules is \kappa -accessible.

$\mathcal P_1$-covers over commutative rings

In this paper we consider the class \mathcal P_1(R) of modules of projective dimension at most one over a commutative ring R and we investigate when \mathcal P_1(R) is a covering class. More precisely, we investigate Enochs' Conjecture, that is the question of whether \mathcal P_1(R) is covering necessarily implies that \mathcal P_1(R) is closed under direct limits. We answer the question affirmatively in the case of a commutative semihereditary ring R . This gives an example of a cotorsion pair (\mathcal P_1(R), \mathcal P_1(R)^\perp) which is not necessarily of finite type such that \mathcal P_1(R) satisfies Enochs' Conjecture. Moreover, we describe the class \varinjlim \mathcal P_1(R) over (not necessarily commutative) rings which admit a classical ring of quotients.

The Homotopy Category of Cotorsion Flat Modules

This paper aims at studying the homotopy category of cotorsion flat left modules $${{\mathbb {K}}({\mathrm{CotF}}\text {-}R)}$$ over a ring R. We prove that if R is right coherent, then the homotopy category $${\mathbb {K}}(\mathrm{dg}\text {-}\mathrm{CotF}\text {-}R)$$ of dg-cotorsion complexes of flat R-modules is compactly generated. This uses firstly the existence of cotorsion flat preenvelopes over such rings and, secondly, the existence of a complete cotorsion pair $$({{\mathbb {K}}_{\mathrm{p}}({\mathrm{Flat}}\text {-}R)}, {\mathbb {K}}(\mathrm{dg}\text {-}\mathrm{CotF}\text {-}R))$$ in the homotopy category $${{\mathbb {K}}({\mathrm{Flat}}\text {-}R)}$$ of complexes of flat R-modules, for arbitrary R. In the setting of quasi coherent sheaves over a Noetherian scheme, this cotorsion pair was discovered in the literature. However, we use a more elementary argument that gives this cotorsion pair for arbitrary R. Next we deal with cotorsion flat resolutions of complexes and define and study the notion of cotorsion flat dimension for complexes of flat R-modules. We also obtain an equivalence $${\mathbb {K}}(\mathrm{dg}\text {-}\mathrm{CotF}\text {-}R)\approx {{\mathbb {K}}({\mathrm{Proj}}\text {-}R)}$$ of triangulated categories where $${{\mathbb {K}}({\mathrm{Proj}}\text {-}R)}$$ is the homotopy category of projective R-modules. Combined with the aforementioned result, this recovers a result from Neeman, asserting the compact generation of $${{\mathbb {K}}({\mathrm{Proj}}\text {-}R)}$$ over right coherent R. Also we get the unbounded derived category $${\mathbb {D}} (R)$$ of R as a Verdier quotient of $${\mathbb {K}}(\mathrm{dg}\text {-}\mathrm{CotF}\text {-}R)$$ .

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.