This paper mainly studies the relative Gorenstein objects in the extriangulated category $\mathcal{C}=(\mathcal{C},\mathbb{E},\mathfrak{s})$ with a proper class $ξ$ and the related properties of these objects.
In the first part, we define the notion of the $ξ$-$\mathcal{G}$projective resolution, and study the relation between $ξ$-projective resolution and $ξ$-$\mathcal{G}$projective resolution for any object $A$ in $\mathcal{C}$, i.e. $A$ has a $\mathcal{C}(-,\mathcal{P}(ξ))$-exact $ξ$-projective resolution if and only if $A$ has a $\mathcal{C}(-,\mathcal{P}(ξ))$-exact $ξ$-$\mathcal{G}$projective resolution.
In the second part, we define a particular $ξ$-Gorenstein projective object in $\mathcal{C}$ which called $ξ$-$n$-strongly $\mathcal{G}$projective object. On this basis, we study the relation between $ξ$-$m$-strongly $\mathcal{G}$projective object and $ξ$-$n$-strongly $\mathcal{G}$projective object whenever $m\neq n$, and give some equivalent characterizations of $ξ$-$n$-strongly $\mathcal{G}$projective objects. What is more, we give some nice propsitions of $ξ$-$n$-strongly $\mathcal{G}$projective objects.
Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category with a proper class $ξ$ of $\mathbb{E}$-triangles. In this paper, we study Gorenstein derived functors for extriangulated categories. More precisely, we first introduce the notion of the proper $ξ$-Gorenstein projective resolution for any object in $\mathcal{C}$ and define the functors $ξ\text{xt}_{\mathcal{GP}(ξ)}$ and $ξ\text{xt}_{\mathcal{GI}(ξ)}$. Under some assumptions, we give some equivalent characterizations for any object with finite $ξ$-Gorenstein projective dimension. Next we get some nice results by using der…
Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category with a proper class $ξ$ of $\mathbb{E}$-triangles. In this paper, we first introduce the $ξ$-Gorenstein cohomology in terms of $ξ$-$\mathcal{G}$projective resolutions and $ξ$-$\mathcal{G}$injective coresolutions, respectively, and then we get the balance of $ξ$-Gorenstein cohomology. Moreover, we study the interplay among $ξ$-cohomology, $ξ$-Gorenstein cohomology and $ξ$-complete cohomology, and obtain the Avramov-Martsinkovsky type exact sequences in this setting.
Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category with a proper class $ξ$ of $\mathbb{E}$-triangles, and $\mathcal{W}$ an additive full subcategory of $\mathcal C$. We provide a method for constructing a proper $\mathcal{W}(ξ)$-resolution (respectively, coproper $\mathcal{W}(ξ)$-coresolution) of one term in an $\mathbb{E}$-triangle in $ξ$ from that of the other two terms. By using this way, we establish the stability of the Gorenstein category $\mathcal{GW}(ξ)$ in extriangulated categories. These results generalise their work by Huang and Yang-Wang, but the proof is not…
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…
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…
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.
Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category with a proper class $ξ$ of $\mathbb{E}$-triangles. The authors introduced and studied $ξ$-$\mathcal{G}$projective and $ξ$-$\mathcal{G}$injective in \cite{HZZ}. In this paper, we discuss Gorenstein homological dimensions for extriangulated categories. More precisely, we first give some characterizations of $ξ$-$\mathcal{G}$projective dimension by using derived functors on $\mathcal{C}$. Second, let $\mathcal{P}(ξ)$ (resp. $\mathcal{I}(ξ)$) be a generating (resp. cogenerating) subcategory of $\mathcal{C}$. We show that the…
Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category with a proper class $ξ$ of $\mathbb{E}$-triangles. In this paper, we study the balance of complete cohomology in $(\mathcal{C},\mathbb{E},\mathfrak{s})$, which is motivated by a result of Nucinkis that complete cohomology of modules is not balanced in the way the absolute cohomology Ext is balanced. As an application, we give some criteria for identifying a triangulated catgory to be Gorenstein and an artin algebra to be $F$-Gorenstein.
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 C be triangulated category and X a cluster tilting subcategory of C. Koenig and Zhu showed that the quotient category C/X is Gorenstein of Gorenstein dimension at most one. The notion of an extriangulated category was introduced by Nakaoka and Palu as a simultaneous generalization of exact categories and triangulated categories. Now let C be extriangulated category with enough projectives and enough injectives, and X a cluster tilting subcategory of C. In this article, we show that under certain conditions the quotient category C/X is Gorenstein of Gorenstein dimension at most one. As an a…
We give a characterisation of the extriangulated categories which admit the structure of a triangulated category. We show that these are the extriangulated categories where for every object $X$ in the extriangulated category, the morphism $0 \rightarrow X$ is a deflation and the morphism $X \rightarrow 0$ is an inflation.
Extriangulated categories were introduced by Nakaoka and Palu, which is a simultaneous generalization of exact categories and triangulated categories. The axiom (ET4) for extriangulated categories is an analogue of the octahedron axiom (TR4) for triangulated categories. In this paper, we introduce homotopy cartesian squares in pre-extriangulated categories to investigate the axiom (ET4). We provide several equivalent statements of the axiom (ET4) and find out conditions under which the axiom is self-dual.
Right triangulated categories can be thought of as triangulated categories whose shift functor is not an equivalence. We give intrinsic characterisations of when such categories have a natural extriangulated structure and are appearing as the (co-)aisle of a (co-)t-structure in an associated triangulated category.
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.
Relative theories(=closed subfunctors) are considered in exact, triangulated and extriangulated categories by Dräxler-Reiten-Smalø-Solberg-Keller, Beligiannis and Herschend-Liu-Nakaoka, respectively. We give a construction method of closed subfunctors from given half exact functors which contains existing constructions. Moreover, if an extriangulated category has enough projective objects, then every closed subfunctor is obtained by this construction.
In this article, we prove that if $(\mathcal A ,\mathcal B,\mathcal C)$ is a recollement of extriangulated categories, then torsion pairs in $\mathcal A$ and $\mathcal C$ can induce torsion pairs in $\mathcal B$, and the converse holds under natural assumptions. Besides, we give mild conditions on a cluster tilting subcategory on the middle category of a recollement of extriangulated categories, for the corresponding abelian quotients to form a recollement of abelian categories.
Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category. Given a composition of two commutative squares in $\mathcal{C}$, if two commutative squares are homotopy cartesian, then their composition is also a homotopy cartesian. This covers the result by Mac Lane (1998) for abelian categories and the result by Christensen and Frankland (2022) for triangulated categories.
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).
Recently, Nakaoka and Palu introduced a notion of extriangulated categories. This is a unification of exact categories and triangulated categories. In this paper, we generalize the definitions of Hall algebras of exact categories and triangulated categories to extriangulated categories.
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…
Presilting and silting subcategories in extriangulated categories were introduced by Adachi and Tsukamoto recently. In this paper, we prove that the Gabriel-Zisman localization $\mathcal B/({\rm thick}\mathcal W)$ of an extriangulated category $\mathcal B$ with respect to a presilting subcategory $\mathcal W$ satisfying certain condition can be realized as a subfactor category of $\mathcal B$. This generalizes the result by Iyama-Yang for silting reduction on triangulated categories. Then we discuss the relation between silting subcategories and tilting subcategories in extriangulated categori…
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…
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.
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.
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）
Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category with a proper class $ξ$ of $\mathbb{E}$-triangles. In this paper, we study complete cohomology of objects in $(\mathcal{C},\mathbb{E},\mathfrak{s})$ by applying $ξ$-projective resolutions and $ξ$-injective coresolutions constructed in $(\mathcal{C},\mathbb{E},\mathfrak{s})$. Vanishing of complete cohomology detects objects with finite $ξ$-projective dimension and finite $ξ$-injective dimension. As a consequence, we obtain some criteria for the validity of the Wakamatsu Tilting Conjecture and give a necessary and sufficie…
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）
Extriangulated categories axiomatize extension-closed subcategories of triangulated categories. We show that the homotopy category of an exact quasi-category can be equipped with a natural extriangulated structure.
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.
In this article, we initiate the study of hereditary extriangulated categories. Many important categories arising in representation theory in connection with various theories of mutation are hereditary extriangulated. Special cases include homotopy categories of 2-term complexes with projective components, which are related to silting mutation, and cluster categories (with relevant relative extriangulated structures) where cluster tilting mutation take place. We prove that there is a theory of irreducible mutation for maximal rigid objects and subcategories in hereditary extriangulated categor…
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…
In this paper, we study a close relationship between relative cluster tilting theory in extriangulated categories and tau-tilting theory in module categories. Our main results show that relative rigid objects are in bijection with $τ$-rigid pairs, and also relative maximal rigid objects with support tau-tilting pairs under some assumptions. These results generalize their work by Adachi-Iyama-Reiten, Yang-Zhu and Fu-Geng-Liu. Finally, we introduce mutation of relative maximal rigid objects and show that any basic relative almost maximal rigid object has exactly two non-isomorphic indecomposable…
A notion of balanced pairs in an extriangulated category with a negative first extension is defined in this article. We prove that there exists a bijective correspondence between balanced pairs and proper classes $ξ$ with enough $ξ$-projectives and enough $ξ$-injectives. It can be regarded as a simultaneous generalization of Fu-Hu-Zhang-Zhu and Wang-Li-Huang. Besides, we show that if $(\mathcal A ,\mathcal B,\mathcal C)$ is a recollement of extriangulated categories, then balanced pairs in $\mathcal B$ can induce balanced pairs in $\mathcal A$ and $\mathcal C$ under natural assumptions. As a a…
We give a classification of substructures (= closed subbifunctors) of a given skeletally small extriangulated category by using the category of defects, in a similar way to the author's classification of exact structures of a given additive category. More precisely, for an extriangulated category, possible substructures are in bijection with Serre subcategories of an abelian category consisting of defects of conflations. As a byproduct, we prove that for a given skeletally small additive category, the poset of exact structures on it is isomorphic to the poset of Serre subcategories of some abe…
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,…
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…
We study the Grothendieck monoid (a monoid version of the Grothendieck group) of an extriangulated category, and give some results which are new even for abelian categories. First, we classify Serre subcategories and dense 2-out-of-3 subcategories using the Grothendieck monoid. Second, in good situations, we show that the Grothendieck monoid of the localization of an extriangulated category is isomorphic to the natural quotient monoid of the original Grothendieck monoid. This includes the cases of the Serre quotient of an abelian category and the Verdier quotient of a triangulated category. As…
We initiate the study of derived functors in the setting of extriangulated categories. By using coends, we adapt Yoneda's theory of higher extensions to this framework. We show that, when there are enough projectives or enough injectives, thus defined extensions agree with the ones defined earlier via projective or injective resolutions. For categories with enough projective or enough injective morphisms, we prove that these are right derived functors of the $\operatorname{Hom}$-bifunctor in either argument. Since $\operatorname{Hom}$ is only half-exact in each argument, it is natural to expec…
