Let $\mathsf{T}$ be a triangulated category with shift functor $Σ\colon \mathsf{T} \to \mathsf{T}$. Suppose $(\mathsf{A},\mathsf{B})$ is a co-t-structure with coheart $\mathsf{S} = Σ\mathsf{A} \cap \mathsf{B}$ and extended coheart $\mathsf{C} = Σ^2 \mathsf{A} \cap \mathsf{B} = \mathsf{S} * Σ\mathsf{S}$, which is an extriangulated category. We show that there is a bijection between co-t-structures $(\mathsf{A}',\mathsf{B}')$ in $\mathsf{T}$ such that $\mathsf{A} \subseteq \mathsf{A}' \subseteq Σ\mathsf{A}$ and complete cotorsion pairs in the extended coheart $\mathsf{C}$. In the case that $\mathsf{T}$ is Hom-finite, $\mathbf{k}$-linear and Krull-Schmidt, we show further that there is a bijection between complete cotorsion pairs in $\mathsf{C}$ and functorially finite torsion pairs in $\mathsf{mod}\, \mathsf{S}$.
In this paper we study cotorsion and torsion pairs induced by cotilting modules. We prove the existence of a strong relationship between the $Σ$-pure injectivity of the cotilting module and the property of the induced cotorsion pair to be of finite type. In particular for cotilting modules of injective dimension at most 1, or for noetherian rings, the two notions are equivalent. On the other hand we prove that a torsion pair is cogenerated by a $Σ$-pure injective cotilting module if and only if its heart is a locally noetherian Grothendieck category. Moreover we prove that any ring admitting a…
We develop theory of (possibly large) cotilting objects of injective dimension at most one in general Grothendieck categories. We show that such cotilting objects are always pure-injective and that they characterize the situation where the Grothendieck category is tilted using a torsion pair to another Grothendieck category. We prove that for Noetherian schemes with an ample family of line bundles a cotilting class is closed under injective envelopes if and only if it is invariant under twists by line bundles, and that such cotilting classes are parametrized by specialization closed subsets di…
In this work, we revisit Auslander-Buchweitz Approximation Theory and find some relations with cotorsion pairs and model category structures. From the notions of relatives generators and cogenerators in Approximation Theory, we introduce the concept of left Frobenius pairs $(\mathcal{X},ω)$ in an abelian category $\mathcal{C}$. We show how to construct from $(\mathcal{X},ω)$ a projective exact model structure on $\mathcal{X}^\wedge$, as a result of Hovey-Gillespie Correspondence applied to two compatible and complete cotorsion pairs in $\mathcal{X}^\wedge$. These pairs can be regarded as examp…
Let $\mathcal B$ be an extriangulated category with enough projectives and enough injectives. We define a proper $m$-term subcategory $\mathcal G$ on $\mathcal B$, which is an extriangulated subcategory. Then we give a correspondence between cotorsion pairs on $\mathcal G$, support $τ$-tilting subcategories on an abelian quotient of $\mathcal G$ when $m=2$. If such $\mathcal G$ is induced by a hereditary cotorsion pair, then we give a correspondence between cotorsion pairs on $\mathcal G$ and intermediate cotorsion pairs on $\mathcal B$ under certain assumptions. At last, we study an important…
In this article, we investigate the condition for the hearts of twin cotorsion pairs to be equivalent, compatibly with the associated functors. This is related to the vanishing of components of pairs through the associated functors.
We show a cotorsion pair cogenerated by a class is complete under suitable conditions in an arbitrary exact category using the generalized small object argument given by Chorny. This recovers Saorín and Šťovíček's criterion of the completeness of cotorsion pairs in their efficient exact categories. Examples in the categories of chain complexes of exact categories are given.
We present the concept of cotorsion pairs cut along subcategories of an abelian category. This provides a generalization of complete cotorsion pairs, and represents a general framework to find approximations restricted to certain subcategories. We also exhibit some connections between cut cotorsion pairs and Auslander-Buchweitz approximation theory, by considering relative analogs for Frobenius pairs and Auslander-Buchweitz contexts. Several applications are given in the settings of relative Gorenstein homological algebra, chain complexes and quasi-coherent sheaves, but also to characterize so…
Given a complete hereditary cotorsion pair $(\mathcal{A}, \mathcal{B})$ in an abelian category $\mathcal{C}$ satisfying certain conditions, we study the completeness of the induced cotorsion pairs $(Φ(\mathcal{A}), Φ(\mathcal{A})^{\perp})$ and $(^{\perp}Ψ(\mathcal{B}), Ψ(\mathcal{B}) )$ in the category $\mbox{Rep}(Q, \mathcal{C})$ of $\mathcal{C}$-valued representations of a given quiver $Q$. We show that if $Q$ is left rooted, then the cotorsion pair $(Φ(\mathcal{A}), Φ(\mathcal{A})^{\perp})$ is complete, and if $Q$ is right rooted, then the cotorsion pair $(^{\perp}Ψ(\mathcal{B}), Ψ(\mathcal…
In this article, we study localizations of hearts of cotorsion pairs (U,V) where U is rigid on an extriangulated category B. The hearts of such cotorsion pairs are equivalent to the functor categories over the stable category of U. Inspired by Marsh and Palu, we consider the mutation of U that induces a cotorsion pair (U',V'). Generally speaking, the hearts of (U,V) and (U',V') are not equivalent to each other, but we will give a generalized pseudo-Morita equivalence between certain localizations of their hearts.
A notion of $n$-cotorsion pairs in an extriangulated category with enough projectives and enough injectives is defined in this article. We show that there exists a one-to-one correspondence between $n$-cotorsion pairs and $(n+1)$-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.
Nakaoka and Palu introduced the notion of extriangulated categories by extracting the similarities between exact categories and triangulated categories. In this paper, we study cotorsion pairs in a Frobenius extriangulated category $\C$. Especially, for a $2$-Calabi-Yau extriangulated category $\C$ with a cluster structure, we describe the cluster substructure in the cotorsion pairs. For rooted cluster algebras arising from $\C$ with cluster tilting objects, we give a one-to-one correspondence between cotorsion pairs in $\C$ and certain pairs of their rooted cluster subalgebras which we call c…
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 kind of hearts of twin cotorsion pairs induced by $d$-cluster tilting subcategories in extriangulated categories. We give a necessary and sufficient condition for such hearts to be abelian. In particular, we also can see that such hearts are hereditary. As an application, this generalizes the work by Liu in an exact case, thereby providing new insights in a triangulated case.
Model structure arising from one hereditary complete cotorsion pair on extriangulated categories
School of Mathematics, Hangzhou Normal University（杭州师范大学数学学院）；School of Mathematical Sciences, Zhejiang Normal University（浙江师范大学数学科学学院）；School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）；School of Mathematics and Statistics, Changsha University of Science and Technology（长沙理工大学数学与统计学院）
In this paper, we first construct some complete cotorson pairs on the category $\mathbb{C}_N(\mathcal{G})$ of unbounded $N$-complexes of Grothendieck category $\mathcal{G}$, from two given cotorsion pairs in $\mathcal{G}$. Next as an application, we focus on particular homotopy categories and the existence of adjoint functors between them. These are an $N$-complex version of the results were shown by Neeman in the category of ordinary complexes.
We give a simultaneous generalization of exact categories and triangulated categories, which is suitable for considering cotorsion pairs, and which we call extriangulated categories. Extension-closed, full subcategories of triangulated categories are examples of extriangulated categories. We give a bijective correspondence between some pairs of cotorsion pairs which we call Hovey twin cotorsion pairs, and admissible model structures. As a consequence, these model structures relate certain localizations with certain ideal quotients, via the homotopy category which can be given a triangulated st…
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 $\mathc…
In the papers of Nakaoka, he introduced the notion of hearts of (twin) cotorsion pairs on triangulated categories and showed that they have structures of (semi-) abelian categories. We study in this article a twin cotorsion pair (S,T),(U,V) on an exact category B with enough projectives and injectives and introduce a notion of the heart. First we show that its heart is preabelian. Moreover we show the heart of a single cotorsion pair is abelian. These results are analog of Nakaoka's results in triangulated categories. We also consider special cases where the heart has nicer structure. By our r…
In this article, we study the heart of a cotorsion pairs on an exact category and a triangulated category in a unified meathod, by means of the notion of an extriangulated category. We prove that the heart is abelian, and construct a cohomological functor to the heart. If the extriangulated category has enough projectives, this functor gives an equivalence between the heart and the category of coherent functors over the coheart modulo projectives. We also show how an n-cluster tilting subcategory of an extriangulated category gives rise to a family of cotorsion pairs with equivalent hearts.
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 Chen-Liu-Yang in a triangulated case. Moreover, it highlights new phenomena when it applied to an exact category. Finally, we give some applications to illustrate our main results. In particular, we obtain the Krause's recollement whose the proofs are both elementary and very general.
It was shown recently that the heart of a twin cotorsion pair ((S,T),(U,V)) 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 T=U, then we show that the Gabriel-Zisman localisation of the heart at the class of its regular morphisms is equivalent to the heart of the single twin cotorsion pair (S,T). This generalises and improves the known result for triangulated categories, thereby pr…
Assume that $(\mathcal{C}, \mathbb{E}, \mathfrak{s})$ is an extriangulated category satisfying Condition (WIC). Let $(\cal Q, \widetilde{\cal R})$ and $(\widetilde{\cal Q}, \cal R)$ be two hereditary cotorsion pairs satisfying conditions $\widetilde{\cal R} \subseteq \cal R$, $\widetilde{\cal Q}\subseteq \cal Q$ and $\widetilde{\cal Q}\cap \cal R = \cal Q \cap \widetilde{\cal R}$. Then there exists a unique thick class $\cal W$ for which $(\cal Q,\cal W,\cal R)$ is a Hovey triple. As an application, this result generalizes the work by Gillespie in an exact case. Moreover, it highlights new phe…
Extended heart construction (I): The heart of $n$-cotorsion pairs on triangulated categories
Graduate School of Mathematics, Nagoya University（名古屋大学大学院数学研究科）；Department of Mathematical Sciences, University of the Ryukyus（冲绳大学数学科学系）；Faculty of Engineering Science, Kansai University（关西大学工程科学部）
Chains of model structures arising from cotorsion pairs on extriangulated categories
School of Mathematical Sciences, Zhejiang University of Technology（浙江工业大学数学科学学院）；School of Science, Zhejiang University of Science and Technology（浙江科技大学理学院）；School of Mathematical Sciences, Zhejiang Normal University（浙江师范大学数学与计算机科学学院）；School of Mathematics and Statistics, Changsha University of Science and Technology（长沙理工大学数学与统计学院）
For a Calabi-Yau triangulated category $\mathcal{C}$ of Calabi-Yau dimension $d$ with a $d-$cluster tilting subcategory $\mathcal{T}$, it is proved that the decomposition of $\mathcal{C}$ is determined by the special decomposition of $\mathcal{T}$, namely, $\mathcal{C}=\oplus_{i\in I}\mathcal{C}_i$, where $\mathcal{C}_i, i\in I$ are triangulated subcategories, if and only if $\mathcal{T}=\oplus_{i\in I}\mathcal{T}_i,$ where $\mathcal{T}_i, i\in I$ are subcategories with $Hom_{\mathcal{C}}(\mathcal{T}_i[t],\mathcal{T}_j)=0, \forall 1\leq t\leq d-2$ and $i\not= j.$ This induces that the Gabriel …
Salce \cite{MR565595} introduced the notion of a \emph{cotorsion pair} of classes of abelian groups, and asked whether every such pair is \emph{complete} (i.e., has enough injectives and projectives). We prove that it is consistent, relative to the consistency of Vopěnka's Principle (VP), that the answer is affirmative. Combined with a previous result of Eklof-Shelah \cite{MR2031314}, this shows that Salce's Problem is independent of the ZFC axioms (modulo the consistency of VP).
In this article, we introduce the notion of {\it concentric twin cotorsion pair} on a triangulated category. This notion contains the notions of $t$-structure, cluster tilting subcategory, co-$t$-structure and functorally finite rigid subcategory as examples. Moreover, a recollement of triangulated categories can be regarded as a special case of concentric twin cotorsion pair. To any concentric twin cotorsion pair, we associate a pretriangulated subquotient category. This enables us to give a simultaneous generalization of the Iyama-Yoshino reduction and the recollement of cotorsion pairs. Thi…
In this paper, we introduce and study relative phantom morphisms in extriangulated categories defined by Nakaoka and Palu. Then using their properties, we show that if $(\C,\E,\s)$ is an extriangulated category with enough injective objects and projective objects, then there exists a bijective correspondence between any two of the following classes: (1) special precovering ideals of $\C$; (2) special preenveloping ideals of $\C$; (3) additive subfunctors of $\E$ having enough special injective morphisms; and (4) additive subfunctors of $\E$ having enough special projective morphisms. Moreover,…
Let $\mathscr{C}$ be an extriangulated category with a proper class $ξ$ of $\mathbb{E}$-triangles. We introduce the notions of left Frobenius pairs, left ($n$-)cotorsion pairs and left (weak) Auslander-Buchweitz contexts with respect to $ξ$ in $\mathscr{C}$. We show how to construct left cotorsion pais from left $n$-cotorsion pairs, and establish a one-to-one correspondence between left Frobenius pairs and left (weak) Auslander-Buchweitz contexts. We also study the relation between a certain class of cotorsion pairs and that of $n$-cotorsion pairs. These work generalize Ma-Zhao-Huang's results…
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 $(Φ(\mathcal{A}_{A},\mathcal{C}_{B}), \mathrm{Rep}(\mathcal{B}_{A},\mathcal{D}_{B}))$ and $(\mathrm{Rep}(\mathcal{A}_{A},\mathcal{C}_{B}), Ψ(\mathcal{B}_{A},\mathcal{D}_{B}))$ in $T$-Mod and show that both of these cotorsion pairs are complete and hereditary. Given two cofibrantly ge…
We study the category $\mathrm{Rep}(Q,\mathcal{M})$ of representations of a quiver $Q$ with values in an abelian category $\mathcal{M}$. Under certain assumptions, we show that every cotorsion pair $(\mathcal{A},\mathcal{B})$ in $\mathcal{M}$ induces two (explicitly described) cotorsion pairs $(Φ(\mathcal{A}),\mathrm{Rep}(Q,\mathcal{B}))$ and $(\mathrm{Rep}(Q,\mathcal{A}),Ψ(\mathcal{B}))$ in $\mathrm{Rep}(Q,\mathcal{M})$. This is akin to a result by Gillespie, which asserts that a cotorsion pair $(\mathcal{A},\mathcal{B})$ in $\mathcal{M}$ induces cotorsion pairs $(\widetilde{\mathcal{A}}, \ma…
Let $F:\mathcal{A}\to \mathcal{B}$ be a left adjoint between abelian categories and let $Ch(F)$ be the induced left adjoint on chain complexes. If the abelian categories $\mathcal{A}$ and $\mathcal{B}$ are equipped with sufficiently nice cotorsion pairs, then we find model structures on their categories of chain complexes. This paper gives novel criteria under which the induced adjunction becomes a Quillen adjunction. In particular, our criteria can be checked on the level of the underlying abelian categories instead of chain complexes. We do the same for adjunctions of $n$ variables and we al…
We study the interplay between the notions of $n$-coherent rings and finitely $n$-presented modules, and also study the relative homological algebra associated to them. We show that the $n$-coherency of a ring is equivalent to the thickness of the class of finitely $n$-presented modules. The relative homological algebra part comes from the study of orthogonal complements to this class of modules with respect to ${\rm Ext}^1_R(F,-)$ and ${\rm Tor}_1^R(F,-)$. We also construct cotorsion pairs from these orthogonal complements, allowing us to provide further characterizations of $n$-coherent ring…
Cotorsion pairs and model structures induced by resolution dimensions relative to Frobenius pairs
Centro de Ciencias Matemáticas. Universidad Nacional Autónoma de México（墨西哥国立自治大学数学科学中心）；Instituto de Matemática y Estadística “Prof. Ing. Rafael Laguardia”. Facultad de Ingeniería. Universidad de la República（乌拉圭共和国大学工程学院拉富尔加里亚教授数学与统计研究所）
Given a complete hereditary cotorsion pair $(\mathcal{A},\mathcal{B})$ in a Grothendieck category $\mathcal{G}$, the derived category $\mathcal{D}(\mathcal{B})$ of the exact category $\mathcal{B}$ is defined as the quotient of the category $\mathrm{Ch}(\mathcal{B})$, of unbounded complexes with terms in $\mathcal{B}$, modulo the subcategory $\widetilde{\mathcal{B}}$ consisting of the acyclic complexes with terms in $\mathcal{B}$ and cycles in $\mathcal{B}$. We restrict our attention to the cotorsion pairs such that $\widetilde{\mathcal{B}}$ coincides with the class $ex\mathcal{B}$ of the acycl…
We study ideal cotorsion pairs associated to weak proper classes of triangles in extension closed subcategories of triangulated categories. This approach allows us to extend the recent ideal approximations theory developed by Fu, Herzog et al. for exact categories in the above mentioned context, and to provide simplified proofs for the ideal versions of some standard results as Salce's Lemma, Wakamatsu's Lemma and Christensen's Ghost Lemma. In the last part of the paper we apply the theory in order to study connections between projective classes (in particular localization or smashing subcateg…
