arXiv subjects

Panyue Zhou

Publications and source records attributed to Panyue Zhou.

At least 19 recordsLinked to original sources

arXiv2026

The Auslander-Reiten conjecture for algebras with radical cube zero

Let $A$ be a split finite-dimensional algebra over a field whose radical $J$ satisfies $J^3=0$, and let $s$ be the number of isomorphism classes of simple $A$-modules. We prove that a non-projective module $M$ with ${\rm Ext}_A^i(M,A)=0$ for all $i>0$ has a non-zero self-extension in some degree between $1$ and $3s+1$. In particular, $A$ satisfies the Auslander--Reiten conjecture, which asserts that every self-orthogonal generator is projective. As a consequence, every finite-dimensional algebra over an algebraically closed field with radical cube zero satisfies the Auslander-Reiten conjecture.

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arXiv2026

A negative answer to a question on tilting objects and two-term complexes

Let $M$ be a silting object in an idempotent complete algebraic triangulated category $\mathcal T$. Put $B={\rm End}_{\mathcal T}(M)$, and let $\mathbb{P}_M\colon {\rm pr}(M)\to K^{[-1,0]}({\rm proj}B)$ be the presentation functor associated with $M$. It was recently asked whether $\mathbb{P}_M(T)$ must be tilting whenever $T\in{\rm pr}(M)$ is a tilting object. We answer this question in the negative by giving an explicit finite-dimensional example. Namely, for $$ \Lambda=k(1\xrightarrow{\alpha}2\xrightarrow{\beta}3\xrightarrow{\gamma}4)/(\alpha\beta\gamma),$$ we construct a silting object $M\in K^b({\rm proj}\Lambda)$ and a tilting object $T=\Sigma\Lambda\in{\rm pr}(M)$ for which $$ {\rm dim}_k{\rm Hom}_{K^b({\rm proj}B)}\bigl(\mathbb{P}_M(T),\Sigma^{-1}\mathbb{P}_M(T)\bigr)=1.$$ Thus $\mathbb{P}_M(T)$ is a two-term silting complex but not a tilting complex.

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arXiv2026

A right pretriangulated category which is not right triangulated

Chen, Liu, Lu, and Zhang recently constructed a pretriangulated category with invertible suspension in which Verdier's octahedral axiom fails. We introduce a general enlargement construction for right pretriangulated categories and show that it preserves axioms (RTR1)-(RTR3), while failure of (RTR4) is detected by the forgetful functor. Applied to their type $A_5$ example, the construction yields a right pretriangulated category that is not right triangulated. In this example, the suspension is faithful but not essentially surjective.

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arXiv2026

A pre-$(n+2)$-angulated category which is not $(n+2)$-angulated

We construct an explicit pre-$9$-angulated category which is not $9$-angulated, thereby giving a genuinely higher counterexample to the implication from pre-$(n+2)$-angulated to $(n+2)$-angulated. The underlying additive category is the category of finitely generated projective right modules over the preprojective algebra $\Pi(A_5)$ over $\mathbb F_2$. The construction is obtained by taking an odd power of the twisted complete comparison used by Chen-Liu-Lu-Zhang in their pre-triangulated counterexample and by showing that the resulting pre-$9$-angulation fails the higher mapping-cone axiom.

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arXiv2026

Auslander-Reiten (n+2)-angles and local finiteness

Let $\mathcal C$ be an $(n+2)$-angulated category. Zhou proved that, when $n$ is odd, if the Auslander-Reiten $(n+2)$-angles generate the relations for the Grothendieck group of $\mathcal C$, then $\mathcal C$ is locally finite. Whether the corresponding statement remains valid for even $n$ is still open. In this paper, we give a partial affirmative answer to this problem by establishing a sufficient condition under which the same implication holds for even $n$. We further show that our sufficient condition is satisfied by a broad class of examples, thereby demonstrating that the result extends well beyond isolated cases.

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arXiv2026

Ideal $n$-cotorsion pairs in Frobenius extriangulated categories

Motivated by the correspondence between ideal cotorsion pairs in Frobenius exact categories and those in their stable categories, we introduce the notion of an ideal $n$-cotorsion pair in an extriangulated category. We study the relationship between ideal $n$-cotorsion pairs in a Frobenius extriangulated category $\mathcal C$ and those in its stable category $\underline{\mathcal C}=\mathcal C/\omega$. Our main result shows that $(\mathcal I,\mathcal J)$ is an ideal $n$-cotorsion pair in $\mathcal C$ if and only if $(\mathcal I/\omega,\mathcal J/\omega)$ is an ideal $n$-cotorsion pair in $\underline{\mathcal C}$. This provides a bridge between higher ideal approximation theory in Frobenius extriangulated categories and its counterpart in their stable categories. Additionally, in Krull--Schmidt exact categories, we establish a bijective correspondence between complete cotorsion pairs and complete ideal cotorsion pairs, answering a question of Fu, Guil Asensio, Herzog and Torrecillas.

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arXiv2026

Quasi-abelian quotients in extriangulated categories

Let $(\mathcal{E}, \mathbb{E}, \mathfrak{s})$ be an extriangulated category. Motivated by the theory of hereditary algebras, we introduce the notion of a hereditary-type subcategory $\mathcal{W}\subseteq \mathcal{E}$. We prove that the quotient $\mathcal{E}/\mathcal{W}$ is a quasi-abelian category, that is, an additive category with kernels and cokernels in which kernels are stable under pushouts and cokernels are stable under pullbacks. Moreover, we show that $\mathcal{E}/\mathcal{W}$ is abelian if and only if $\mathcal{W}$ is a cluster tilting subcategory in a suitable relative extriangulated structure. Several examples are provided to illustrate the main results, showing that our approach both recovers known abelian hearts and yields new abelian or quasi-abelian quotients beyond classical settings.

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arXiv2026

Chains of model structures arising from cotorsion pairs on extriangulated categories

The main aim of this paper is to study chains of model structures arising from cotorsion pairs in extriangulated categories. Starting with a hereditary Hovey triple, we construct further hereditary Hovey triples whose homotopy categories are equivalent under suitable completeness assumptions, thereby refining results due to El Maaouy and Shao-Wang-Zhang. As an application, we consider objects of finite Gorenstein injective dimension with respect to a proper class of $\mathbb{E}$-triangles. Under mild set-theoretic assumptions, we obtain a chain of model structures whose homotopy categories are all triangulated equivalent to a common stable category. This recovers known results for Gorenstein injective modules and yields new examples in the derived category of a ring when the proper class is given by cohomological ghost triangles.

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arXiv2025

Triangulated categories arising from n-fold matrix factorizations

Let $\mathcal{A}$ be an additive category and let $T\colon \mathcal{A}\rightarrow \mathcal{A}$ be an additive functor equipped with a natural transformation $\omega\colon \mathrm{Id}_{\mathcal{A}}\rightarrow T$. We prove that the homotopy category of $n$-fold matrix factorizations of $\omega$, denoted ${\rm HFact}_{n}(\mathcal{A},T,\omega)$, admits a natural structure of a right triangulated category. In particular, when $T$ is an automorphism, the homotopy category ${\rm HFact}_{n}(\mathcal{A},T,\omega)$ becomes triangulated. Furthermore, if $\mathcal{A}$ is a Frobenius exact category and $T$ is an autoequivalence, we obtain that the category ${\rm Fact}_{n}(\mathcal{A},T,\omega)$ of $n$-fold $(\mathcal{A},T)$-factorizations of $\omega$ is a Frobenius exact category. Consequently, the stable category of the Frobenius exact category ${\rm Fact}_{n}(\mathcal{A},T,\omega)$ is a triangulated category.

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arXiv2025

Grothendieck groups of repetitive cluster categories

In order to study cluster-tilted algebras and their intermediate coverings, Zhu introduced the notion of repetitive cluster categories, defined as the orbit categories $\mathcal D^b(\mathcal H)/\langle(τ^{-1}Σ)^p\rangle$ for $1\leq p\in\mathbb{N}$, where $\mathcal H$ is a hereditary abelian category with tilting objects. In this paper, we compute partial but essential results on the Grothendieck groups of the repetitive cluster categories $\mathcal D^b({\rm mod}KA_n)/\langle(τ^{-1}Σ)^p\rangle$ and $\mathcal D^b({\rm mod} KD_n)/\langle(τ^{-1}Σ)^p\rangle$. Our results extend the known computations for classical cluster categories, reveal new structural patterns arising from the repetitive parameter $p$, and provide further evidence of the close interplay between Grothendieck groups, Auslander-Reiten theory, and Coxeter transformations.

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arXiv2025

$n$-cotorsion pairs in a recollement of extriangulated categories

Let $(\mathcal{A}, \mathcal{B}, \mathcal{C})$ be a recollement of extriangulated categories.In this paper, we first show how to obtain an $n$-cotorsion pair in $\mathcal{B}$ from given $n$-cotorsion pairs in $\mathcal{A}$ and $\mathcal{C}$. Conversely, we prove that an $n$-cotorsion pair in $\mathcal{B}$ can induce $n$-cotorsion pairs in $\mathcal{A}$ and $\mathcal{C}$ under suitable conditions. As applications, several related results are provided to illustrate our construction.

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arXiv2025

Mutation of $n$-cotorsion pairs in extriangulated categories

In this article, we introduce the notion of $n$-cotorsion pairs in extriangulated categories, which extends both the cotorsion pairs established by Nakaoka and Palu and the $n$-cotorsion pairs in triangulated categories developed by Chang and Zhou. We further prove that any mutation of an $n$-cotorsion pair remains an $n$-cotorsion pair. As applications, we provide a geometric characterization of $n$-cotorsion pairs in $n$-cluster categories of type $A_{\infty}$, and we realize mutations of $n$-cotorsion pairs geometrically via rotations of certain configurations of $n$-admissible arcs.

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arXiv2025

Extriangulated length categories: torsion classes and $τ$-tilting theory

This paper introduces the notion of extriangulated length categories, whose prototypical examples include abelian length categories and bounded derived categories of finite dimensional algebras with finite global dimension. We prove that an extriangulated category $\mathcal{A}$ is a length category if and only if $\mathcal{A}$ admits a simple-minded system. Subsequently, we study the partially ordered set ${\rm tor}_Θ(\mathcal{A})$ of torsion classes in an extriangulated length category $(\mathcal{A},Θ)$ from the perspective of lattice theory. It is shown that ${\rm tor}_Θ(\mathcal{A})$ forms a complete lattice, which is further proved to be completely semidistributive and algebraic. Moreover, we describe the arrows in the Hasse quiver of ${\rm tor}_Θ(\mathcal{A})$ using brick labeling. Finally, we introduce the concepts of support torsion classes and support $τ$-tilting subcategories in extriangulated length categories and establish a bijection between these two notions, thereby generalizing the Adachi-Iyama-Reiten bijection for functorially finite torsion classes.

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arXiv2025

Normed modules and the categorification of integrations, series expansions, and differentiations

We explore the assignment of norms to $\mathitΛ$-modules over a finite-dimensional algebra $\mathitΛ$, resulting in the establishment of normed $\mathitΛ$-modules. Our primary contribution lies in constructing two new categories $\mathscr{N}\!\!or^p$ and $\mathscr{A}^p$, where each object in $\mathscr{N}\!\!or^p$ is a normed $\mathitΛ$-module $N$ limited by a special element $v_N\in N$ and a special $\mathitΛ$-homomorphism $δ_N: N^{\oplus 2^{\dim\mathitΛ}} \to N$, the morphism in $\mathscr{N}\!\!or^p$ is a $\mathitΛ$-homomorphism $θ: N\to M$ such that $θ(v_N) = v_M$ and $θδ_N = δ_Mθ^{\oplus 2^{\dim\mathitΛ}}$, and $\mathscr{A}^p$ is a full subcategory of $\mathscr{N}\!\!or^p$ generated by all Banach modules. 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 $\mathscr{A}^p$.

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arXiv2024

Cotorsion pairs in $(d+2)$-angulated categories

Let $\mathcal C$ be a $(d+2)$-angulated category. In this paper, we define the notions of cotorsion pairs and weak cotorsion pairs in $\mathcal C$, which are generalizations of the classical cotorsion pairs in triangulated categories. As an application, we give a geometric characterization of weak cotorsion pairs in $(d+2)$-angulated cluster categories of type $A$. Moreover, we prove that any mutation of a (weak) cotorsion pair in $\mathcal C$ is again a (weak) cotorsion pair. When $d=1$, this result generalizes the work of Zhou and Zhu on classical cotorsion pairs in triangulated categories.

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arXiv2024

On endomorphism algebras of string almost gentle algebras

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.

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arXiv2024

Classifying Nichols algebras over classical Weyl groups

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.

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arXiv2024

Higher Auslander-Reiten sequences revisited

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.

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