Cotorsion Pairs Research Articles

Enochs conjecture for cotorsion pairs over recollements of exact categories

Abstract Let ( A , C , ℬ ) \left({\mathcal{A}},{\mathcal{C}},{\mathcal{ {\mathcal B} }}) be a recollement of exact categories. An explicit procedure about gluing complete hereditary cotorsion pairs from A {\mathcal{A}} and ℬ {\mathcal{ {\mathcal B} }} to C {\mathcal{C}} has been established by Hu et al. to provide a new method on the construction of recollements of triangulated categories. In this article, we study when the validity of Enochs conjecture for the left-hand classes of those complete hereditary cotorsion pairs is preserved in the aforementioned gluing procedure. Applications are given to cotorsion pairs induced by the class of projective objects or Gorenstein projective objects over comma categories.

Dualizations of Approximations, ℵ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\aleph _1$$\\end{document}-Projectivity, and Vopěnka’s Principles

The approximation classes of modules that arise as components of cotorsion pairs are tied up by Salce’s duality. Here we consider general approximation classes of modules and investigate possibilities of dualization in dependence on closure properties of these classes. While some proofs are easily dualized, other dualizations require large cardinal principles, and some fail in ZFC, with counterexamples provided by classes of ℵ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\aleph _1$$\\end{document}-projective modules over non-perfect rings. For example, we show that the statement “each covering class of modules closed under homomorphic images is of the form Gen(M)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathrm{Gen\\,}}(M)$$\\end{document} for a module M” is equivalent to Vopěnka’s Principle.

Compatible weak factorization systems and model structures

In this paper, the concept of compatible weak factorization systems in general categories is introduced as a counterpart of compatible complete cotorsion pairs in abelian categories. We describe a method to construct model structures on general categories via two compatible weak factorization systems satisfying certain conditions, and hence, generalize a very useful result by Gillespie for abelian model structures. As particular examples, we show that weak factorization systems associated to some classical model structures (for example, the Kan-Quillen model structure on sSet) satisfy these conditions.

Cotorsion pairs and model structures on Morita rings

We study cotorsion pairs and abelian model structures on Morita rings Λ=(ANBAMABB) which are Artin algebras. Given cotorsion pairs (U,X) and (V,Y) in A-Mod and B-Mod, respectively, one can construct four cotorsion pairs in Λ-Mod:((XY)⊥,(XY)),(Δ(U,V),Δ(U,V)⊥),((UV),(UV)⊥),(∇⊥(X,Y),∇(X,Y)). These cotorsion pairs have relations:Δ(U,V)⊥⊆(XY),∇⊥(X,Y)⊆(UV). An important feature is that they are not equal, in general. In fact, there even exists a Morita algebra Λ, such that the four cotorsion pairs are pairwise different. The problem of identifications, i.e., when these inclusions are the same, are studied; the heredity and completeness of these cotorsion pairs are investigated; and finally, various model structures on Λ-Mod are obtained, by explicitly giving the corresponding Hovey triples and Quillen's homotopy categories. In particular, cofibrantly generated Hovey triples, and the Gillespie-Hovey triples induced by compatible generalized projective (respectively, injective) cotorsion pairs, are explicitly constructed. All these Hovey triples obtained are pairwise different and “new” in some sense. Some results are even new when M=0 or N=0.

Dimensions and cotorsion pairs in recollements of extriangulated categories

Let ( A , B , C ) be a recollement of extriangulated categories. In this paper, we provide bounds on the coresolution dimensions of the subcategories involved in A , B , and C . We show that a hereditary cotorsion pair in B can induce hereditary cotorsion pairs in A and C under certain conditions. Besides, we construct recollements of abelian categories from a recollement of extriangulated categories with respect to the hearts of cotorsion pairs.

Categories of quiver representations and relative cotorsion pairs

We study the category Rep(Q,C) of representations of a quiver Q with values in an abelian category C. For this purpose we introduce the mesh and the cone-shape cardinal numbers associated to the quiver Q and we use them to impose conditions on C that allow us to prove interesting homological properties of Rep(Q,C) that can be constructed from C. For example, we compute the global dimension of Rep(Q,C) in terms of the global one of C. We also review a result of H. Holm and P. Jørgensen which states that (under certain conditions on C) every hereditary complete cotorsion pair (A,B) in C induces the hereditary complete cotorsion pairs (Rep(Q,A),Rep(Q,A)⊥1) and (Ψ⊥1(B),Ψ(B)) in Rep(Q,C), and then we obtain a strengthened version of this and other related results. Finally, we will apply the above developed theory to study the following full abelian subcategories of Rep(Q,C), finite-support, finite-bottom-support and finite-top-support representations. We show that the above mentioned cotorsion pairs in Rep(Q,C) can be restricted nicely on the aforementioned subcategories and under mild conditions we also get hereditary complete cotorsion pairs.

Completeness of Induced Cotorsion Pairs in Representation Categories of Rooted Quivers

Completeness of Induced Cotorsion Pairs in Representation Categories of Rooted Quivers

Relative tilting theory in abelian categories I: Auslander–Buchweitz–Reiten approximations theory in subcategories and cotorsion-like pairs

In this paper, we introduce a special kind of relative (co)resolutions associated with a pair of classes of objects in an abelian category C.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {C}.$$\\end{document} We will see that, by studying this relative (co)resolutions, we get a possible generalization of a part of the Auslander–Buchweitz approximation theory that is useful for developing n-X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {X}$$\\end{document}-tilting theory in Argudin Monroy and Mendoza Hernández (relative tilting theory in abelian categories II: n-X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {X}$$\\end{document} tilting theory. arXiv:2112.14873, 2021). With this goal, new concepts as X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {X}$$\\end{document}-complete and X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {X}$$\\end{document}-hereditary pairs are introduced as a generalization of complete and hereditary cotorsion pairs. These pairs appear in a natural way in the study of the category of representations of a quiver in an abelian category (Argudin Monroy and Mendoza Hernández in categories of quiver representations and relative cotorsion pairs. arXiv:2311.12774v1, 2023). Our main results will include an existence theorem for relative approximations, among other results related with closure properties of relative (co)resolution classes and relative homological dimensions which are essential in the development of n-X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {X}$$\\end{document}-tilting theory in Argudin Monroy and Mendoza Hernández (2021).

Homotopy in Exact Categories

In this monograph we develop various aspects of the homotopy theory of exact categories. We introduce different notions of compactness and generation in exact categories, and use these to study model structures on categories of chain complexes C h ∗ ( E ) Ch_{*}(\mathcal {E}) which are induced by cotorsion pairs on E \mathcal {E} . As a special case we show that under very general conditions the categories C h + ( E ) Ch_{+}(\mathcal {E}) , C h ≥ 0 ( E ) Ch_{\ge 0}(\mathcal {E}) , and C h ( E ) Ch(\mathcal {E}) are equipped with the projective model structure, and that a generalisation of the Dold-Kan correspondence holds. We also establish conditions under which categories of filtered objects in exact categories are equipped with natural model structures. When E \mathcal {E} is monoidal we also examine when these model structures are monoidal and conclude by studying some homotopical algebra in such categories. In particular we provide conditions under which C h ( E ) Ch(\mathcal {E}) and C h ≥ 0 ( E ) Ch_{\ge 0}(\mathcal {E}) are homotopical algebra contexts, thus making them suitable settings for derived geometry.

Left Frobenius Pairs, Cotorsion Pairs and Weak Auslander-Buchweitz Contexts in Triangulated Categories

Let [Formula: see text] be a triangulated category with a proper class [Formula: see text] of triangles. We introduce the notions of left Frobenius pairs, left ([Formula: see text]-)cotorsion pairs and left (weak) Auslander-Buchweitz contexts with respect to [Formula: see text] in [Formula: see text]. We show how to construct left cotorsion pais from left [Formula: see text]-cotorsion pairs, and establish a one-to-one correspondence between left Frobenius pairs and left (weak) Auslander-Buchweitz contexts. Some applications are given in the Gorenstein homological theory of triangulated categories.

Largest exact structures and almost split sequences on hearts of twin cotorsion pairs

Abstract Hearts of cotorsion pairs on extriangulated categories are abelian categories. On the other hand, hearts of twin cotorsion pairs are not always abelian. They were shown to be semi-abelian by Liu and Nakaoka. Moreover, Hassoun and Shah proved that they are quasi-abelian under certain conditions. In this article, we first show that the heart of any twin cotorsion pair has a largest exact category structure and is always quasi-abelian. We also provide a sufficient and necessary condition for the heart of a twin cotorsion pair being abelian. Then by using the results we have got, we investigate the almost split sequences in the hearts of twin cotorsion pairs. Finally, as an application, we show that a Krull–Schmidt, Hom-finite triangulated category has a Serre functor whenever it has a cluster tilting object.

Hearts of twin cotorsion pairs revisited: Integral and Abelian hearts

Hearts of twin cotorsion pairs are shown to be quasi-abelian in [14]. But they are not always integral. In this article, we provide a sufficient and necessary condition for the hearts of twin cotorsion pairs being integral (resp. abelian).

Enochs’ conjecture for cotorsion pairs and more

Abstract Enochs’ conjecture asserts that each covering class of modules (over any ring) has to be closed under direct limits. Although various special cases of the conjecture have been verified, the conjecture remains open in its full generality. In this paper, we prove the conjecture for the classes Filt ⁡ ( 𝒮 ) {\operatorname{Filt}(\mathcal{S})} , where 𝒮 {\mathcal{S}} consists of ℵ n {\aleph_{n}} -presented modules for some fixed n < ω {n<\omega} . In particular, this applies to the left-hand class of any cotorsion pair generated by a class of ℵ n {\aleph_{n}} -presented modules. Moreover, we also show that it is consistent with ZFC that Enochs’ conjecture holds for all classes of the form Filt ⁡ ( 𝒮 ) {\operatorname{Filt}(\mathcal{S})} , where 𝒮 {\mathcal{S}} is a set of modules. This leaves us with no explicit example of a covering class where we cannot prove that Enochs’ conjecture holds (possibly under some additional set-theoretic assumption).

Silting objects and recollements of extriangulated categories

<p>In this paper, let $ (\mathscr{A}, \mathscr{C}, \mathscr{B}) $ be a recollement of extriangulated categories. We construct a silting object and a tilting object from the two end terms of a recollement. We also show that the reverse direction holds under natural assumptions. Moreover, we show that our gluing preserves cotorsion pairs.</p>

Separated monic correspondence of cotorsion pairs and semi-Gorenstein-projective modules

Given a finite dimensional algebra A over a field k, and a finite acyclic quiver Q, let Λ=A⊗kkQ/I, where kQ is the path algebra of Q over k and I is a monomial ideal. This paper is devoted to studying the separated monic representations over the bounded quiver (Q,I) over a subcategory X of A-mod, denoted by smon(Q,I,X). We show that (X,Y) is a (complete) hereditary cotorsion pair in A-mod if and only if (smon(Q,I,X),rep(Q,I,Y)) is a (complete) hereditary cotorsion pair in Λ-mod. We also show that A is left weakly Gorenstein if and only if so is Λ. Provided that kQ/I is non-semisimple, the category ▪ of semi-Gorenstein-projective Λ-modules coincides with the category of separated monic representations ▪ if and only if A is left weakly Gorenstein.

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. In this article, we consider a special class of hearts of twin cotorsion pairs induced by [Formula: see text]-cluster tilting subcategories in extriangulated categories. We give a necessary and sufficient condition for such hearts to be abelian. In particular, we can also see that such hearts are hereditary. As an application, this generalizes the work by Liu in the exact case, thereby providing new insights into the triangulated case.

Auslander conditions and tilting-like cotorsion pairs

We study homological behavior of modules satisfying the Auslander condition. Assume that AC is the class of left R-modules satisfying the Auslander condition. It is proved that each cycle of an exact complex with each term in AC belongs to AC for any ring R. As a consequence, we show that for any left Noetherian ring R, AC is a resolving subcategory of the category of left R-modules if and only if RR satisfies the Auslander condition if and only if each Gorenstein projective left R-module belongs to AC. As an application, we prove that, for an Artinian algebra R satisfying the Auslander condition, R is Gorenstein if and only if AC coincides with the class of Gorenstein projective left R-modules if and only if (AC<∞,(AC<∞)⊥) is a tilting-like cotorsion pair if and only if (AC<∞,I) is a tilting-like cotorsion pair, where AC<∞ is the class of left R-modules with finite AC-dimension and I is the class of injective left R-modules. This leads to some criteria for the validity of the Auslander and Reiten conjecture which says that an Artinian algebra satisfying the Auslander condition is Gorenstein.

Cotorsion pairs and Hovey triples over trivial ring extensions

Let be a trivial extension of a ring R by an R-R-bimodule M. We first give some homological formulas over . Then we investigate how to construct (hereditary, complete) cotorsion pairs and (hereditary, cofibrantly generated) Hovey triples over . Finally, some applications are given in some Morita context rings.

Generalized tilting theory in functor categories

AbstractThis paper is devoted to the study of generalized tilting theory of functor categories in different levels. First, we extend Miyashita’s proof (Math Z 193:113–146,1986) of the generalized Brenner–Butler theorem to arbitrary functor categories $\mathop{\textrm{Mod}}\nolimits\!(\mathcal{C})$ with $\mathcal{C}$ an annuli variety. Second, a hereditary and complete cotorsion pair generated by a generalized tilting subcategory $\mathcal{T}$ of $\mathop{\textrm{Mod}}\nolimits \!(\mathcal{C})$ is constructed. Some applications of these two results include the equivalence of Grothendieck groups $K_0(\mathcal{C})$ and $K_0(\mathcal{T})$ , the existences of a new abelian model structure on the category of complexes $\mathop{\textrm{C}}\nolimits \!(\!\mathop{\textrm{Mod}}\nolimits\!(\mathcal{C}))$ , and a t-structure on the derived category $\mathop{\textrm{D}}\nolimits \!(\!\mathop{\textrm{Mod}}\nolimits \!(\mathcal{C}))$ .

Weakly Ding injective modules and complexes

We consider two classes of modules over coherent rings: the Gorenstein FP-injective modules, and the weakly Ding injective modules. The Gorenstein FP-injective modules are the cycles of the exact complexes of FP-injective modules. The weakly Ding injective modules are the cycles of the exact complexes of FP-injective modules that stay exact when applying a functor for any FP-injective module A. We prove that the class of Gorenstein FP-injective modules is both covering and preenveloping over any (left) coherent ring with the property that every injective module has finite flat dimension. We also prove that, over the same type of rings, the class of weakly Ding injectives, , is preenveloping in . If, moreover, is closed under extensions, then is a hereditary cotorsion pair. In particular, this is the case when R is a Ding Chen ring, and is closed under extensions. We show that if R is a Ding Chen ring such that every FP-injective module has finite projective dimension, then the two classes of modules coincide. Consequently, the class of weakly Ding injective modules is also covering in this case. In the last part of the paper we prove some analogue results for Gorenstein FP-injective complexes and for weakly Ding injective complexes.