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For any arbitrary string almost gentle algebra, we consider specific subsets of its quiver's arrow set, denoted by $\mathcal{R}$. For each such $\mathcal{R}$, we introduce the finitely generated module $M_{\mathcal{R}}$ and define its associated $\mathcal{R}$-endomorphism algebra $A_{\mathcal{R}}$. In this paper, we show that the representation type of a string gentle algebra $A$, the representation type of the $\mathcal{R}$-endomorphism algebra $A_{\mathcal{R}}$ for some $\mathcal{R}$, the representation types of all $\mathcal{R}$-algebras, and the representation type of the Cohen-Macaulay Auslander algebra $A^{\mathrm{CMA}}$ of $A$ are equivalent. The results presented here reveal a deep structural connection between different classes of algebras derived from string gentle algebras. By showing the equivalence of representation types, this work offers new insights into the nature of endomorphism algebras and Cohen-Macaulay Auslander algebras, contributing to a broader understanding of their algebraic properties and classification., Comment: 22 pages, 8 figures

In this article, we show that conjugacy classes of classical Weyl groups $W(B_{n})$ and $W(D_{n})$ are of $\textit{type D}$. Consequently, we obtain that Nichols algebras of irreducible Yetter-Drinfeld modules over the classical Weyl groups $\mathbb W_{n}$ ($n\geq5$) are infinite dimensional., Comment: 7 pages

Let $(\mathscr{C},\mathbb{E},\mathfrak{s})$ be an $n$-exangulated category with enough projectives and enough injectives, and $\mathscr{X}$ be a cluster-tilting subcategory of $\mathscr{C}$. Liu and Zhou have shown that the quotient category $\mathscr{C}/\mathscr{X}$ is an $n$-abelian category. In this paper, we prove that if $\mathscr{C}$ has Auslander-Reiten $n$-exangles, then $\mathscr{C}/\mathscr{X}$ has Auslander-Reiten $n$-exact sequences. Moreover, we also show that if a Frobenius $n$-exangulated category $\mathscr{C}$ has Auslander-Reiten $n$-exangles, then the stable category $\overline{\mathscr{C}}$ of $\mathscr{C}$ has Auslander-Reiten $(n+2)$-angles., Comment: 19 pages. arXiv admin note: text overlap with arXiv:2006.02223, arXiv:2108.07985, arXiv:2110.02476

Extriangulated categories, introduced by Nakaoka and Palu, serve as a simultaneous generalization of exact and triangulated categories. In this paper, we first introduce the concept of admissible weak factorization systems and establish a bijection between cotorsion pairs and admissible weak factorization systems in extriangulated categories. Consequently, we give the equivalences between hereditary cotorsion pairs and compatible cotorsion pairs via admissible weak factorization systems under certain conditions in extriangulated categories, thereby generalizing a result by Di, Li, and Liang., Comment: 13 pages

Let $\mathcal{C}$ be a weakly idempotent complete extriangulated category. In contrast with the Hovey correspondence of admissible model structures on weakly idempotent complete exact categories from two complete cotorsion pairs, we give a construction of model structures on $\mathcal{C}$ from only one complete cotorsion pair. Our main result not only generalizes the work by Beligiannis-Reiten and Cui-Lu-Zhang, but also provides methods to construct model structures from silting objects of $\mathcal{C}$ and co-$t$-structures in triangulated categories., Comment: 23 pages

We explore the assignment of norms to $\Lambda$-modules over a finite-dimensional algebra $\Lambda$, resulting in the establishment of normed $\Lambda$-modules. Our primary contribution lies in constructing a new category $\mathscr{N}\!\!or^p$ related to normed modules along with its full subcategory $\mathscr{A}^p$. By examining the objects and morphisms in these categories, we establish a framework for understanding the categorification of integration, series expansions and derivatives. Furthermore, we obtain the Stone--Weierstrass approximation theorem in the sense of $\mathscr{A}^p$., Comment: 36 pages, 1 figures

Let $\mathcal C$ be a Krull-Schmidt triangulated category with shift functor $[1]$ and $\mathcal R$ be a rigid subcategory of $\mathcal C$. We are concerned with the mutation of two-term weak $\mathcal R[1]$-cluster tilting subcategories. We show that any almost complete two-term weak $\mathcal R[1]$-cluster tilting subcategory has exactly two completions. Then we apply the results on relative cluster tilting subcategories to the domain of $\tau$-tilting theory in functor categories and abelian categories., Comment: 35 pages.In this version, we have added many new results

In this article, we define the notion of $n$-cotorsion pairs in triangulated categories, which is a generalization of the classical cotorsion pairs. We prove that any mutation of an $n$-cotorsion pair is again an $n$-cotorsion pair. When $n=1$, this result generalizes the work of Zhou and Zhu for classical cotorsion pairs. As applications, we give a geometric characterization of $n$-cotorsion pairs in $n$-cluster categories of type $A$ and give a geometric realization of mutation of $n$-cotorsion pairs via rotation of certain configurations of $n$-diagonals., Comment: 13 pages

Herschend-Liu-Nakaoka introduced the concept of $n$-exangulated categories as higher-dimensional analogues of extriangulated categories defined by Nakaoka-Palu. The class of $n$-exangulated categories contains $n$-exact categories and $(n+2)$-angulated categories as specific examples. In this article, we introduce the notion of hereditary $n$-exangulated categories, which generalize hereditary extriangulated categories. We provide two classes of hereditary $n$-exangulated categories through closed subfunctors. Additionally, we define the concept of $0$-Auslander $n$-exangulated categories and discuss the circumstances under which these two classes of hereditary $n$-exangulated categories become $0$-Auslander., Comment: 15 pages. arXiv admin note: substantial text overlap with arXiv:2109.12954, arXiv:2006.02223, arXiv:2108.07985, arXiv:2111.06522, arXiv:1909.13284

Let $(\mathcal A, \mathcal B, \mathcal C)$ be a recollement of extriangulated categories. In this paper, we provide bounds on the coresolution dimensions of the subcategories involved in $\mathcal A$, $\mathcal B$ and $\mathcal C$. We show that a hereditary cotorsion pair in $\mathcal B$ can induce hereditary cotorsion pairs in $\mathcal A$ and $\mathcal C$ under certain conditions. Besides, we construct recollements of abelian categories from a recollement of extriangulated categories with respect to the hearts of cotorsion pairs., Comment: 19 pages

Hearts of twin cotorsion pairs are shown to be quasi-abelian. 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)., Comment: 15 pages

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., Comment: We have changed the title and added Sections 5 and 6

Let $(\mathcal B,\mathbb{E},\mathfrak{s})$ be an extriangulated category and $\mathcal S$ be an extension closed subcategory of $\mathcal B$. In this article, we prove that the Gabriel-Zisman localization $\mathcal B/\mathcal S$ can be realized as an ideal quotient inside $\mathcal B$ when $\mathcal S$ satisfies some mild conditions. The ideal quotient is an extriangulated category. We show that the equivalence between the ideal quotient and the localization preserves the extriangulated category structure. We also discuss the relations of our results with Hovey twin cotorsion pairs and Verdier quotients., Comment: 26 pages.This article has been accepted for publication in SCIENCE CHINA Mathematics

In this article, we introduce the notion of pre-$(n+2)$-angulated categories as higher dimensional analogues of pre-triangulated categories defined by Beligiannis-Reiten. We first show that the idempotent completion of a pre-$(n+2)$-angulated category admits a unique structure of pre-$(n+2)$-angulated category. Let $(\mathscr{C},\mathbb{E},\mathfrak{s})$ be an $n$-exangulated category and $\mathscr{X}$ be a strongly functorially finite subcategory of $\mathscr{C}$. We then show that the quotient category $\mathscr{C}/\mathscr{X}$ is a pre-$(n+2)$-angulated category.These results allow to construct several examples of pre-$(n+2)$-angulated categories. Moreover, we also give a necessary and sufficient condition for the quotient $\mathscr{C}/\mathscr{X}$ to be an $(n+2)$-angulated category., Comment: 20 pages. arXiv admin note: text overlap with arXiv:2108.07985, arXiv:2006.02223

It was shown recently that an $n$-extension closed subcategory $\mathscr A$ of a Krull-Schmidt $(n+2)$-angulated category has a natural structure of an $n$-exangulated category. In this article, we prove that its idempotent completion $\widetilde{\mathscr A}$ admits an $n$-exangulated structure. It is not only a generalization of the main result of Lin, but also gives an $n$-exangulated category which is neither $n$-exact nor $(n+2)$-angulated in general., Comment: 11 pages. arXiv admin note: text overlap with arXiv:2206.09658. substantial text overlap with arXiv:2109.02068; text overlap with arXiv:2011.00729, arXiv:2109.12954, arXiv:2108.07985, arXiv:2006.02223

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., Comment: 8 pages

$n$-exangulated categories were introduced by Herschend-Liu-Nakaoka which are a simultaneous generalization of $n$-exact categories and $(n+2)$-angulated categories. This paper consists of two results on $n$-exangulated categories: (1) we give an equivalent characterization of the axiom (EA2); (2) we provide a new way to construct a closed subfunctor of an $n$-exangulated category., Comment: 11 pages

Nakaoka-Ogawa-Sakai considered the localization of an extriangulated category. This construction unified the Serre quotient of abelian categories and the Verdier quotient of triangulated categories. Recently, Herschend-Liu-Nakaoka defined $n$-exangulated categories as a higher dimensional analogue of extriangulated categories. Let $\mathcal C$ be an $n$-exangulated category and $\mathcal{F}$ be a multiplicative system satisfying mild assumption. In this article, we give a necessary and sufficient condition for the localization of $\mathcal C$ be an $n$-exangulated category. This way gives a new class of $n$-exangulated categories which are neither $n$-exact nor $(n+2)$-angulated in general. Moreover, our result also generalizes work by Nakaoka-Ogawa-Sakai., Comment: 20 pages. arXiv admin note: text overlap with arXiv:2103.16907, arXiv:1709.06689 by other authors

Let $(\mathscr{C},\mathbb{E},\mathfrak{s})$ be an Ext-finite, Krull-Schmidt and $k$-linear $n$-exangulated category with $k$ a commutative artinian ring. In this note, we prove that $\mathscr{C}$ has Auslander-Reiten-Serre duality if and only if $\mathscr{C}$ has Auslander-Reiten $n$-exangles. Moreover, we also give an equivalent condition for the existence of Serre duality (which is a special type of Auslander-Reiten-Serre duality). Finally, assume further that $\mathscr{C}$ has Auslander-Reiten-Serre duality. We exploit a bijection triangle, which involves the restricted Auslander bijection and the Auslander-Reiten-Serre duality., Comment: 24 pages. arXiv admin note: substantial text overlap with arXiv:2111.06522; text overlap with arXiv:2110.02476

Let $(\mathscr{C},\mathbb{E},\mathfrak{s})$ be an ${\rm Ext}$-finite, Krull-Schmidt and $k$-linear $n$-exangulated category with $k$ a commutative artinian ring. In this note, we define two additive subcategories $\mathscr{C}_r$ and $\mathscr{C}_l$ of $\mathscr{C}$ in terms of the representable functors from the stable category of $\mathscr{C}$ to the category of finitely generated $k$-modules. Moreover, we show that there exists an equivalence between the stable categories of these two full subcategories. Finally, we give some equivalent characterizations on the existence of Auslander-Reiten $n$-exangles via determined morphisms. These results unify and extend their works by Jiao-Le for exact categories, Zhao-Tan-Huang for extriangulated categories, Xie-Liu-Yang for $n$-abelian categories., Comment: 28 pages. arXiv admin note: text overlap with arXiv:2110.02476

Let $\mathscr{C}$ be an $n$-exangulated category. In this note, we show that if $\mathscr{C}$ is locally finite, then $\mathscr{C}$ has Auslander-Reiten $n$-exangles. This unifies and extends results of Xiao-Zhu, Zhu-Zhuang, Zhou and Xie-Lu-Wang for triangulated, extriangulated, $(n+2)$-angulated and $n$-abelian categories, respectively., Comment: 18 pages. arXiv admin note: substantial text overlap with arXiv:2109.12954; text overlap with arXiv:2108.07985, arXiv:2109.00196. text overlap with arXiv:2006.02223, arXiv:2011.00729, arXiv:1909.13284; text overlap with arXiv:1709.06689 by other authors

Let $\mathscr{C}$ be a Krull-Schmidt $n$-exangulated category and $\mathscr{A}$ be an $n$-extension closed subcategory of $\mathscr{C}$. Then $\mathscr{A}$ inherits the $n$-exangulated structure from the given $n$-exangulated category in a natural way. This construction gives $n$-exangulated categories which are not $n$-exact categories in the sense of Jasso nor $(n+2)$-angulated categories in the sense of Geiss-Keller-Oppermann in general. Furthermore, we also give a sufficient condition on when an $n$-exangulated category $\mathscr{A}$ is an $n$-exact category. These results generalize work by Klapproth and Zhou., Comment: 16 pages. arXiv admin note: substantial text overlap with arXiv:2109.02068; text overlap with arXiv:2108.07985, arXiv:2109.00196

Let $\mathscr C$ be a Krull-Schmidt $(n+2)$-angulated category and $\mathscr A$ be an $n$-extension closed subcategory of $\mathscr C$. Then $\mathscr A$ has the structure of an $n$-exangulated category in the sense of Herschend-Liu-Nakaoka. This construction gives $n$-exangulated categories which are not $n$-exact categories in the sense of Jasso nor $(n+2)$-angulated categories in the sense of Geiss-Keller-Oppermann in general. As an application, our result can lead to a recent main result of Klapproth., Comment: 8 pages. arXiv admin note: text overlap with arXiv:2108.07985, arXiv:2109.00196

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 $\xi$ with enough $\xi$-projectives and enough $\xi$-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 application, this result gengralizes a result by Fu-Hu-Yao in abelian categories. Moreover, it highlights a new phenomena when it applied to triangulated categories., Comment: 18 pages. arXiv admin note: text overlap with arXiv:2109.00932. text overlap with arXiv:2104.04924

Herschend-Liu-Nakaoka introduced the notion of $n$-exangulated categories. It is not only a higher dimensional analogue of extriangulated categories defined by Nakaoka-Palu, but also gives a simultaneous generalization of $n$-exact categories and $(n+2)$-angulated categories. In this article, we give an $n$-exangulated version of Auslander's defect and Auslander-Reiten duality formula. Moreover, we also give a classification of substructures (=closed subbifunctors) of a given skeletally small $n$-exangulated category by using the category of defects., Comment: 24 pages. arXiv admin note: text overlap with arXiv:1709.06689 by other authors

Let $n$ be an integer greater or equal than $3$. We give a simultaneous generalization of $(n-2)$-exact categories and $n$-angulated categories, and we call it one-sided $n$-suspended categories. One-sided $n$-angulated categories are also examples of one-sided $n$-suspended categories. We provide a general framework for passing from one-sided $n$-suspended categories to one-sided $n$-angulated categories. Besides, we give a method to construct $n$-angulated quotient categories from Frobenius $n$-prile categories. These results generalize their works by Jasso for $n$-exact categories, Lin for $(n+2)$-angulated categories and Li for one-sided suspended categories., Comment: 19 pages

Recently, Wang, Wei and Zhang define the recollement of extriangulated categories, which is a generalization of both recollement of abelian categories and recollement of triangulated categories. For a recollement $(\mathcal A ,\mathcal B,\mathcal C)$ of extriangulated categories, we show that $n$-tilting (resp. $n$-cotilting) subcategories in $\mathcal A$ and $\mathcal C$ can be glued to get $n$-tilting (resp. $n$-cotilting) subcategories in $\mathcal B$ under certain conditions., Comment: 12 pages

The notion of right semi-equivalence in a right $(n+2)$-angulated category is defined in this article. Let $\mathscr C$ be an $n$-exangulated category and $\mathscr X$ is a strongly covariantly finite subcategory of $\mathscr C$. We prove that the standard right $(n+2)$-angulated category $\mathscr C/\mathscr X$ is right semi-equivalence under a natural assumption. As an application, we show that a right $(n+2)$-angulated category has an $n$-exangulated structure if and only if the suspension functor is right semi-equivalence. Besides, we also prove that an $n$-exangulated category $\mathscr C$ has the structure of a right $(n+2)$-angulated category with right semi-equivalence if and only if for any object $X\in\mathscr C$, the morphism $X\to 0$ is a trivial inflation., Comment: 12 pages. arXiv admin note: text overlap with arXiv:2006.02223

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 categories, this gives us a kind of important examples of our results. In particular, for a finite dimensional Gorenstein algebra, we get the relative version of the description of the singularity category due to Happel and Chen-Zhang by this reduction., Comment: 22 pages

We introduce a new concept of s-recollements of extriangulated categories, which generalizes recollements of abelian categories, recollements of triangulated categories, as well as recollements of extriangulated categories. Moreover, some basic properties of s-recollements are presented. We also discuss the behavior of the localization theory on the adjoint pair of exact functors. Finally, we provide a method to obtain a recollement of triangulated categories from s-recollements of extriangulated categories via the localization theory., Comment: 24 pages

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., Comment: 19 pages

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., Comment: 13 pages

Let $R$ be a ring. In this paper, we study the characterization of cosilting modules and establish a relation between cosilting modules and cotilting objects in a Grothendieck category. We proved that each cosilting right $R$-module $T$ can be described as a cotilting object in $\sigma[R/I]$, where $I$ is a right ideal of $R$ determined by $T$ and $\sigma[R/I]$ is the full subcategory of right $R$-modules, consisting of submodules of $R/I$-generated modules. Conversely, under some suitable conditions, if $T$ is a cotilting object in $\sigma[R/I]$, then $T$ is cosilting., Comment: 10 pages

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 $\tau$-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 property of such extriangulated subcategory $\mathcal G$., Comment: 18 pages

Herschend-Liu-Nakaoka introduced the notion of $n$-exangulated categories. It is not only a higher dimensional analogue of extriangulated categories defined by Nakaoka-Palu, but also gives a simultaneous generalization of $(n+2)$-angulated in the sense of Geiss-Keller-Oppermann and $n$-exact categories in the sense of Jasso. In this article, we show that an $n$-exangulated category has the structure of an $(n+2)$-angulated category if and only if for any object $X$ in the category, the morphism $0\to X$ is a trivial deflation and the morphism $X\to 0$ is a trivial inflation., Comment: 17 pages, comments are welcome. arXiv admin note: text overlap with arXiv:2006.02223, arXiv:2007.01716, arXiv:1909.13284

Let $\mathcal A$ be a Hom-finite abelian category with enough projectives. In this note, we show that any covariantly finite $\tau$-rigid subcategory is contained in a support $\tau$-tilting subcategory. We also show that support $\tau$-tilting subcategories are in bijection with certain finitely generated torsion classes. Some applications of our main results are also given., Comment: 7 pages, comments are welcome

Let $\mathcal B$ be an extriangulated category with enough projectives $\mathcal P$ and enough injectives $\mathcal I$, and let $\mathcal R$ be a contravariantly finite rigid subcategory of $\mathcal B$ which contains $\mathcal P$. We have an abelian quotient category $\mathcal H/\mathcal R\subseteq \mathcal B/\mathcal R$ which is equivalent ${\rm mod}(\mathcal R/\mathcal P)$. In this article, we find a one-to-one correspondence between support $\tau$-tilting (resp. $\tau$-rigid) subcategories of $\mathcal H/\mathcal R$ and maximal relative rigid (resp. relative rigid) subcategories of $\mathcal H$, and show that support tilting subcategories in $\mathcal H/\mathcal R$ is a special kind of support $\tau$-tilting subcategories. We also study the relation between tilting subcategories of $\mathcal B/\mathcal R$ and cluster tilting subcategories of $\mathcal B$ when $\mathcal R$ is cluster tilting., Comment: 21 pages. arXiv admin note: text overlap with arXiv:2003.12788

Herschend-Liu-Nakaoka introduced the notion of $n$-exangulated categories. It is not only a higher dimensional analogue of extriangulated categories defined by Nakaoka-Palu, but also gives a simultaneous generalization of $n$-exact categories and $(n+2)$-angulated categories. Let $\mathcal C$ be an $n$-exangulated category and $\mathcal X$ a full subcategory of $\mathcal C$. If $\mathcal X$ satisfies $\mathcal X\subseteq\mathcal P\cap\mathcal I$, then we give a necessary and sufficient condition for the ideal quotient $\mathcal C/\mathcal X$ to be an $n$-exangulated category, where $\mathcal P$ (resp. $\mathcal I$) is the full subcategory of projective (resp. injective) objects in $\mathcal C$. In addition, we define the notion of $n$-proper class in $\mathcal C$. If $\xi$ is an $n$-proper class in $\mathcal C$, then we prove that $\mathcal C$ admits a new $n$-exangulated structure. These two ways give $n$-exangulated categories which are neither $n$-exact nor $(n+2)$-angulated in general., Comment: We modified Theorem 3.1 and added some examples. arXiv admin note: text overlap with arXiv:1909.13284

We introduce the higher version of the notion of Adachi-Iyama-Reiten's support $\tau$-tilting pairs, which is a generalization of maximal $\tau_n$-rigid pairs in the sense of Jacobsen-J{\o}rgensen. Let $\mathcal C$ be an $(n+2)$-angulated category with an $n$-suspension functor $\Sigma^n$ and an Opperman-Thomas cluster tilting object. We show that relative $n$-rigid objects in $\mathcal C$ are in bijection with $\tau_n$-rigid pairs in the $n$-abelian category $\mathcal C/{\rm add}\Sigma^n T$, and relative maximal $n$-rigid objects in $\mathcal C$ are in bijection with support $\tau_n$-tilting pairs. We also show that relative $n$-self-perpendicular objects are in bijection with maximal $\tau_n$-rigid pairs. These results generalise the work for $\mathcal C$ being $2n$-Calabi-Yau by Jacobsen-J{\o}rgensen and the work for $n=1$ by Yang-Zhu., Comment: 17 pages, comments are welcome

Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category with a proper class $\xi$ of $\mathbb{E}$-triangles. In this paper, we first introduce the $\xi$-Gorenstein cohomology in terms of $\xi$-$\mathcal{G}$projective resolutions and $\xi$-$\mathcal{G}$injective coresolutions, respectively, and then we get the balance of $\xi$-Gorenstein cohomology. Moreover, we study the interplay among $\xi$-cohomology, $\xi$-Gorenstein cohomology and $\xi$-complete cohomology, and obtain the Avramov-Martsinkovsky type exact sequences in this setting., Comment: 18 pages

Herschend-Liu-Nakaoka introduced the notion of $n$-exangulated categories. It is not only a higher dimensional analogue of extriangulated categories defined by Nakaoka-Palu, but also gives a simultaneous generalization of $n$-exact categories in the sense of Jasso and $(n+2)$-angulated in the sense of Geiss-Keller-Oppermann. Let $\mathscr C$ be an $n$-exangulated category with enough projectives and enough injectives, and $\mathscr X$ a cluster tilting subcategory of $\mathscr C$. In this article, we show that the quotient category $\mathscr C/\mathscr X$ is an $n$-abelian category. This extends a result of Zhou-Zhu for $(n+2)$-angulated categories. Moreover, it highlights new phenomena when it is applied to $n$-exact categories., Comment: 18 pages. arXiv admin note: text overlap with arXiv:1909.13284 and arXiv:1807.06733

Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category with a proper class $\xi$ 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., Comment: 18 pages. arXiv admin note: text overlap with arXiv:2003.11852, arXiv:1908.00931

Let B be an extriangulated category with enough projectives and enough injectives. Let C be a fully rigid subcategory of B which admits a twin cotorsion pair ((C,K),(K,D)). The quotient category B/K is abelian, we assume that it is hereditary and has finite length. In this article, we study the relation between support tilting subcategories of B/K and maximal relative rigid subcategories of B. More precisely, we show that the image of any cluster tilting subcategory of B is support tilting in B/K and any support tilting subcategory in B/K can be lifted to a unique relative maximal rigid subcategory in B. We also give a bijection between these two classes of subcategories if C is generated by an object., Comment: 14 pages