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Panyue Zhou

Publications and source records attributed to Panyue Zhou.

At least 73 records · Page 4Linked to original sources
arXiv2020

Hereditary cotorsion pairs on extriangulated subcategories

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 property of such extriangulated subcategory $\mathcal G$.

math.RT↗
arXiv2020

A new characterization of n-exangulated categories with (n+2)-angulated structure

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.

math.CT↗
arXiv2020

$τ$-tilting theory in abelian categories

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

math.RT↗
arXiv2020

Two new classes of n-exangulated categories

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 $ξ$ 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.

math.RT↗
arXiv2020

Relative rigid subcategories and $τ$-tilting theory

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 $τ$-tilting (resp. $τ$-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 $τ$-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.

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arXiv2020

Support $τ_n$-tilting pairs

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

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arXiv2020

Avramov-Martsinkovsky type exact sequences for extriangulated categories

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.

math.CT↗
arXiv2020

From $n$-exangulated categories to $n$-abelian categories

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.

math.RT↗
arXiv2020

Frobenius $n$-exangulated categories

Herschend-Liu-Nakaoka introduced the notion 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 examples. In this article, we introduce a notion of Frobenius $n$-exangulated categories which are a generalization of Frobenius $n$-exact categories. We show that the stable category of a Frobenius $n$-exangulated category is an $(n+2)$-angulated category. As an application, this result generalizes the work by Jasso. We provide a class of $n$-exangulated categories which are neither $n$-exact categories nor $(n+2)$-angulated categories. Finally, we discuss an application of the main results and give some examples illustrating it.

math.RT↗
arXiv2020

Balance of complete cohomology in extriangulated categories

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.

math.CT↗
arXiv2020

Complete cohomology for extriangulated categories

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 sufficient condition for a virtually Gorenstein algebra to be Gorenstein. Moreover, we give a general technique for computing complete cohomology of objects with finite $ξ$-$\mathcal{G}$projective dimension. As an application, the relationships between $ξ$-projective dimensions and $ξ$-$\mathcal{G}$projective dimensions for objects in $(\mathcal{C},\mathbb{E},\mathfrak{s})$ are given.

math.RT↗
arXiv2020

Modules of infinite projective dimension

We characterize the modules of infinite projective dimension over the endomorphism algebras of Opperman-Thomas cluster tilting objects $X$ in $(n+2)$-angulated categories $(\mathcal C,Σ^n,Θ)$. For an indecomposable object $M$ of $\mathcal C$, we define in this article the ideal $I_M$ of ${\rm End}_{\mathcal C}(Σ^nX)$ given by all endomorphisms that factor through ${\rm add} M$, and show that the ${\rm End}_{\mathcal C}(X)$-module ${\rm Hom}_{\mathcal C}(X,M)$ has infinite projective dimension precisely when $I_M$ is non-zero. As an application, we generalize a recent result by Beaudet-Brüstle-Todorov for cluster-tilted algebras.

math.RT↗
arXiv2020

On the relation between relative rigid and support tilting

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.

math.RT↗
arXiv2020

Proper resolutions and Gorensteinness in extriangulated categories

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 too far from their case. Finally, we give some applications about our main results.

math.RT↗
arXiv2020

Proper classes and Gorensteinness in extriangulated categories

Extriangulated categories were introduced by Nakaoka and Palu as a simultaneous generalization of exact categories and triangulated categories. A notion of proper class in an extriangulated category is defined in this paper. Let $\mathcal{C}$ be an extriangulated category and $ξ$ a proper class in $\mathcal{C}$. We prove that $\mathcal{C}$ admits a new extriangulated structure. This construction gives extriangulated categories which are neither exact categories nor triangulated categories. Moreover, we introduce and study $ξ$-Gorenstein projective objects in $\mathcal{C}$ and demonstrate that $ξ$-Gorenstein projective objects share some basic properties with Gorenstein projective objects in module categories or in triangulated categories. In particular, we refine a result of Asadollahi and Salarian [Gorenstein objects in triangulated categories, J. Algebra 281(2004), 264-286]. As an application, the $ξ$-$\mathcal{G}$projective model structures on extriangulated categories are obtained.

math.RT↗
arXiv2020

Grothendieck groups and Auslander-Reiten (d+2)-angles

Xiao and Zhu has shown that if $\cal C$ is a locally finite triangulated category, then the Auslander-Reiten triangles generate the relations for the Grothendieck group of $\cal C$. The notion of $(d +2)$-angulated categories is a "higher dimensional" analogue of triangulated categories. In this article, we show that if A $(d+2)$-angulated category $\cal C$ is locally finite if and only if the Auslander-Reiten $(d+2)$-angles generate the relations for the Grothendieck group of $\cal C$. This extends the result of Xiao and Zhu, and gives the converse of Xiao and Zhu's result is also true.

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