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 providing new insights in the exact setting.
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…
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.
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.
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…
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）
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.
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…
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,…
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）
In this work we introduce notions in Auslander-Buchweitz theory and cotorsion theory in extriangulated categories which extend the given ones for abelian categories. Although these notions have been already developed for extriangulated categories in remarkable works, in this paper, we tackle them in a relative sense by considering subcategories of objects. This approach not only covers the existing theory given on abelian, exact and triangulated categories, but it also shows how to get similar results with an appropriate treatment of local properties.
Extriangulated categories were introduced by Nakaoka and Palu as a simultaneous generalization of exact categories and triangulated categories. In this paper, we show that the idempotent completion of an extriangulated category admits a natural extriangulated structure. As applications, we prove that cotorsion pairs in an extriangulated category induce cotorsion pairs in its idempotent completion under certain condition, and the idempotent completion of a recollement of extriangulated categories is still a recollement.
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…
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）
We give a simultaneous generalization of recollements of abelian categories and triangulated categories, which we call recollements of extriangulated categories. For a recollement $(\mathcal{A}$, $\mathcal{B}$, $\mathcal{C})$ of extriangulated categories, we show that cotorsion pairs in $\mathcal{A}$ and $\mathcal{C}$ induce cotorsion pairs in $\mathcal{B}$ under certain conditions. As an application, our main result recovers a result given by Chen for recollements of triangulated categories, and it also shows a new phenomena when it is applied to abelian categories.
Extriangulated categories were introduced by Nakaoka and Palu as a simultaneous generalization of exact categories and triangulated categories. In this paper, we introduce and develop an analogous theory of Auslander-Buchweitz approximations for extriangulated categories. We establish the existence of precovers pand preenvelopesq and obtain characterizations of relative homological dimensions, which are based on certain subcategories under finiteness of resolutions. Finally, we give a description of cotorsion pairs on extriangulated categories under some conditions, and provide a characterizat…
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.
Extriangulated categories were introduced by Nakaoka and Palu as a simultaneous generalization of exact categories and triangulated categories. In this paper, we first introduce the concept of left Frobenius pairs on an extriangulated category C, and then establish a bijective correspondence between Frobenius pairs and certain cotorsion pairs in C. As an application, some new admissible model structures are established from left Frobenius pairs under certain conditions, which generalizes a result of Hu et al. (J. Algebra 551 (2020) 23-60).
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.
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.
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…
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…
As a general framework for the studies of $t$-structures on triangulated categories and torsion pairs in abelian categories, we introduce the notions of extriangulated categories with negative first extensions and $s$-torsion pairs. We define a heart of an interval in the poset of $s$-torsion pairs, which naturally becomes an extriangulated category with a negative first extension. This notion generalizes hearts of $t$-structures on triangulated categories and hearts of twin torsion pairs in abelian categories. In this paper, we show that an interval in the poset of $s$-torsion pairs is biject…
Let $\mathcal{X}$ be a semibrick in an extriangulated category $\mathscr{C}$. Let $\mathcal{T}$ be the filtration subcategory generated by $\mathcal{X}$. We give a one-to-one correspondence between simple semibricks and length wide subcategories in $\mathscr{C}$. 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 $\mathcal{T}$ and certain subsets of $\mathcal{X}$. Applying to the simple minded systems of an triangulated category, we recover a …
In this article, we show that the localization of an extriangulated category by a multiplicative system satisfying mild assumptions can be equipped with a natural, universal structure of an extriangulated category. This construction unifies the Serre quotient of abelian categories and the Verdier quotient of triangulated categories. Indeed we give such a construction for a bit wider class of morphisms, so that it covers several other localizations appeared in the literature, such as Rump's localization of exact categories by biresolving subcategories, localizations of extriangulated categories…
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 $\mat…
In the paper of Keller and Reiten, it was shown that the quotient of a triangulated category (with some conditions) by a cluster tilting subcategory becomes an abelian category. After that, Koenig and Zhu showed in detail, how the abelian structure is given on this quotient category, in a more abstract setting. On the other hand, as is well known since 1980s, the heart of any $t$-structure is abelian. We unify these two construction by using the notion of a cotorsion pair. To any cotorsion pair in a triangulated category, we can naturally associate an abelian category, which gives back each of…
In this paper, we study the ideal approximation theory associated to almost $n$-exact structures in the $n$-exangulated category. The notions of $n$-ideal cotorsion pairs and $n$-$\mathbb{F}$-phantom morphisms are introduced and studied. In particular, let $\mathscr{C}$ be an extriangulated category which satisfies the condition (WIC) and $\mathcal{T}$ be a nicely embedded $n$-cluster tilting subcategory of $\mathscr{C}$, we prove Salce's Lemma in $\mathcal{T}$.
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 …
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…
