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Zengqiang Lin

Publications and source records attributed to Zengqiang Lin.

17 recordsLinked to original sources
arXiv2025

Intervals of torsion pairs and generalized Happel-Reiten-Smal{\o} tilting

Let $\mathcal{A}$ be an abelian category with a torsion pair $(\mathcal{T},\mathcal{F})$. Happel-Reiten-Smalo tilting provides a method to construct a new abelian category $\mathcal{B}$ with a torsion pair associated to $(\mathcal{T},\mathcal{F})$, which is exactly the heart of a certain $t$-structure on the bounded derived category $D^b(\mathcal{A})$. In this paper, we mainly study generalized HRS tilting. We first show that an interval of torsion pairs in extriangulated categories with negative first extensions is bijectively associated with torsion pairs in the corresponding heart, which yields several new observations in triangulated categories. Then we obtain a generalization of HRS tilting by replacing hearts of $t$-structures with extended hearts. As an application, we show that certain $t$-structures on triangulated subcategories can be extended to $t$-structures on the whole triangulated categories.

math.RT↗
arXiv2024

Affine root systems, stable tubes and a conjecture by Geiss-Leclerc-Schr\"{o}er

Associated to a symmetrisable Cartan matrix $C$, Geiss-Lerclerc-Schr\"{o}er constructed and studied a class of Iwanaga-Gorenstein algebras $H$. They proved a generalised version of Gabriel's Theorem, that is, the rank vectors of $\tau$-locally free $H$-modules are the positive roots of type $C$ when $C$ is of finite type, and conjectured that this is true for any $C$. In this paper, we look into this conjecture when $C$ is of affine type. We construct explicitly stable tubes, some of which have rigid mouth modules, while others not. We deduce that any positive root of type $C$ is the rank vector of some $\tau$-locally free $H$-module. However, the converse is not true in general. Our construction shows that there are $\tau$-locally free $H$-modules whose rank vectors are not roots, when $C$ is of type $\widetilde{\mathbb{B}}_n$, $\widetilde{\mathbb{CD}}_n$, $\widetilde{\mathbb{F}}_{41}$ and $\widetilde{\mathbb{G}}_{21}$, and so the conjecture fails in these four types.

math.RT↗
arXiv2022

The extensions of t-structures

We reformulate a result of Bernhard Keller on extensions of $t$-structures and give a detailed proof. In the study of hereditary $t$-structures, the notions of regular $t$-structures and global dimensions arise naturally.

math.RT↗
arXiv2021

The (ET4) axiom for Extriangulated Categories

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.

math.CT↗
arXiv2021

The Auslander-Reiten quivers of string algebras of affine type $\widetilde{C}$ and a conjecture by Geiss-Leclerc-Schröer

In this paper, we study representations of certain string algebras, which are referred to as of affine type $\widetilde{C}$. We introduce minimal string modules and apply them to explicitly describe components of the Auslander-Reiten quivers of the string algebras and $τ$-locally free modules defined by Geiss-Lerclerc-Schröer. As an application, we prove Geiss-Leclerc-Schröer's conjecture on the correspondence between positive roots of type $\widetilde{C}$ and $τ$-locally free modules of the corresponding string algebras.

math.RT↗
arXiv2020

Abelian quotients arising from extriangulated categories via morphism categories

We investigate abelian quotients arising from extriangulated categories via morphism categories, which is a unified treatment for both exact categories and triangulated categories. Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category with enough projectives $\mathcal{P}$ and $\mathcal{M}$ be a full subcategory of $\mathcal{C}$ containing $\mathcal{P}$. We show that certain quotient category of $\mathfrak{s}\textup{-def}(\mathcal{M})$, the category of $\mathfrak{s}$-deflations $f:M_{1}\rightarrow M_2$ with $M_1,M_2\in\mathcal{M}$, is abelian. Our main theorem has two applications. If $\mathcal{M}=\mathcal{C}$, we obtain that certain ideal quotient category $\mathfrak{s}\textup{-tri}(\mathcal{C})/\mathcal{R}_2$ is equivalent to the category of finitely presented modules $\textup{mod-}\mathcal{C}/[\mathcal{P}]$, where $\mathfrak{s}$-tri$(\mathcal{C})$ is the category of all $\mathfrak{s}$-triangles. If $\mathcal{M}$ is a rigid subcategory, we show that $\mathcal{M}_{L}/[\mathcal{M}]\cong\textup{mod-}(\mathcal{M}/[\mathcal{P}])$ and $\mathcal{M}_{L}/[Ω\mathcal{M}]\cong(\textup{mod-}(\mathcal{M}/[\mathcal{P}])^{\textup{op}})^{\textup{op}}$, where $\mathcal{M}_L$ (resp. $Ω\mathcal{M}$) is the full subcategory of $\mathcal{C}$ of objects $X$ admitting an $\mathfrak{s}$-triangle $\xymatrixrowsep{0.1pc}\xymatrix{X\ar[r]&M_1\ar[r] & M_2\ar@{-->}[r]&} (\textup{resp.} \xymatrixrowsep{0.1pc}\xymatrix{X\ar[r]&P\ar[r] & M\ar@{-->}[r]&})$ with $M_1, M_2\in\mathcal{M}$ (resp. $M\in\mathcal{M}$ and $P\in\mathcal{P}$). In particular, we have $\mathcal{C}/[\mathcal{M}]\cong\textup{mod-}(\mathcal{M}/[\mathcal{P}])$ and $\mathcal{C}/[Ω\mathcal{M}]\cong(\textup{mod-}(\mathcal{M}/[\mathcal{P}])^{\textup{op}})^{\textup{op}}$ provided that $\mathcal{M}$ is a cluster-tilting subcategory.

math.RT↗
arXiv2018

Abelian quotients of the categories of short exact sequences

We mainly investigate abelian quotients of the categories of short exact sequences. The natural framework to consider the question is via identifying quotients of morphism categories as modules categories. These ideas not only can be used to recover the abelian quotients produced by cluster-tilting subcategories of both exact categories and triangulated categories, but also can be used to reach our goal. Let $(\mathcal{C},\mathcal{E})$ be an exact category. We denote by $\mathcal{E}(\mathcal{C})$ the category of bounded complexes whose objects are given by short exact sequences in $\mathcal{E}$ and by $S\mathcal{E}(\mathcal{C})$ the full subcategory formed by split short exact sequences. In general, $\mathcal{E}(\mathcal{C})$ is just an exact category, but the quotient $\mathcal{E}(\mathcal{C})/[S\mathcal{E}(\mathcal{C})]$ turns out to be abelian. In particular, if $(\mathcal{C},\mathcal{E})$ is Frobenius, we present three equivalent abelian quotients of $\mathcal{E}(\mathcal{C})$ and point out that the equivalences are actually given by left and right rotations. The abelian quotient $\mathcal{E}(\mathcal{C})/[S\mathcal{E}(\mathcal{C})]$ admits some nice properties. We explicitly describe the abelian structure, projective objects, injective objects and simple objects, which provide a new viewpoint to understanding Hilton-Rees Theorem and Auslander-Reiten theory. Furthermore, we present some analogous results both for $n$-exact versions and for triangulated versions.

math.RT↗
arXiv2017

A general construction of $n$-angulated categories using periodic injective resolutions

Let $\mathcal{C}$ be an additive category equipped with an automorphism $Σ$. We show how to obtain $n$-angulations of $(\mathcal{C},Σ)$ using some particular periodic injective resolutions. We give necessary and sufficient conditions on $(\mathcal{C},Σ)$ admitting an $n$-angulation. Then we apply these characterizations to explain the standard construction of $n$-angulated categories and the $n$-angulated categories arising from some local rings. Moreover, we obtain a class of new examples of $n$-angulated categories from quasi-periodic selfinjective algebras.

math.RT↗
arXiv2017

Idempotent completion of n-angulated categories

We show that the idempotent completion of an n-angulated category admits a unique n-angulated structure such that the inclusion is an n-angulated functor, which satisfies a universal property.

math.CT↗
arXiv2016

Homotopy cartesian diagrams in n-angulated categories

It has been proved by Bergh and Thaule that the higher mapping cone axiom is equivalent to the higher octahedral axiom for n-angulated categories. In this note, we use homotopy cartesian diagrams to give several new equivalent statements of the higher mapping cone axiom, which are applied to explain the higher octahedral axiom.

math.RT↗
arXiv2015

$n$-angulated categories from self-injective algebras

Let $\mathcal{C}$ be a $k$-linear category with split idempotents, and $Σ:\mathcal{C}\rightarrow\mathcal{C}$ an automorphism. We show that there is an $n$-angulated structure on $(\mathcal{C},Σ)$ under certain conditions. As an application, we obtain a class of examples of $n$-angulated categories from self-injective algebras.

math.RT↗
arXiv2015

Recollement of additive quotient categories

In this note, we define a recollement of additive categories, and prove that such a recollement can induce a recollement of their quotient categories. As an application, we get a recollement of quotient triangulated categories induced by mutation pairs.

math.RT↗
arXiv2014

Right $n$-angulated categories arising from covariantly finite subcategories

We define the notion of right $n$-angulated category, which generalizes the notion of right triangulated category. Let $\mathcal{C}$ be an additive category or $n$-angulated category and $\mathcal{X}$ a covariantly finite subcategory, we show that under certain conditions the quotient $\mathcal{C}/\mathcal{X}$ is a right $n$-angulated category. This result generalizes some previous work.

math.CT↗
arXiv2014

n-angulated quotient categories induced by mutation pairs

We define mutation pair in an n-angulated category and prove that given such a mutation pair, the corresponding quotient category carries a natural n-angulated structure. This result generalizes a theorem of Iyama-Yoshino in classical triangulated category. As an application, we obtain that the quotient category of a Frobenius n-angulated category is also an n-angulated category.

math.CT↗
arXiv2014

Mutation pairs and triangulated quotients

We define mutation pair in a pseudo-triangulated category. We prove that under certain conditions, for a mutation pair in a pseudo-triangulated category, the corresponding quotient category carries a natural triangulated structure. This result unifies many previous constructions of quotient triangulated categories.

math.CT↗
arXiv2012

Nonlinear Lie type Derivations of Von Neumann Algebras

Let $A$ be a von Neumann algebra with no central summands of type $I_1$. We will show that every nonlinear Lie $n$-derivation on $A$ is of the standard form, i.e. it can be expressed as a sum of an additive derivation and a central-valued mapping which annihilates each $(n-1)$-th commutator of $A$.

math.RA↗