arXiv subjects

Dajun Liu

Publications and source records attributed to Dajun Liu.

17 recordsLinked to original sources

arXiv2026

Ideal balanced pairs in n-exangulated categories

This paper is devoted to ideal approximation theory in $n$-exangulated categories. We define ideal balanced pairs and explore their properties and equivalent characterizations. We prove that any complete ideal cotorsion triple $(\mathcal{I},\mathcal{K},\mathcal{J})$ satisfying suitable assumptions yields an ideal balanced pair $(\mathcal{I}, \mathcal{J})$, and the converse holds under additional hypotheses. We further show that for an ideal balanced pair, it is equivalent to an additive subfunctor with enough projective morphisms and enough injective morphisms.

math.CT↗

arXiv2026

Higher Cotorsion Pairs on Image-Nilpotent $η$-Extensions

Let $\cB$ be an abelian category with enough projective and injective objects, and let $ \cA=\cB\ltimes_η\sF $ be an $η$-extension induced by a right exact endofunctor $\sF$. We introduce uniform image-nilpotence for $η$-extensions and characterize it in terms of the nilpotence of the associative natural transformation $η:\sF^2\to\sF$. Using the associated $η$-Loewy filtration, we prove a lifting theorem for right $n$-cotorsion pairs over image-nilpotent $η$-extensions and establish a corresponding completeness theorem. As applications, our results recover and unify related constructions for split nilpotent ring extensions, comma categories, formal triangular matrix rings, Morita context rings and classical trivial extensions.

math.RT↗

arXiv2026

Relative Derived Delooping Levels and $τ$-Tilting Modules

Let $\mathcal{E}$ be an essentially small idempotent-complete exact category with enough projective objects. We develop derived delooping levels in $\mathcal{E}$. We apply this framework to a basic support $τ$-tilting module $T$. We then define the derived delooping level relative to $T$ and compare it with the ordinary relative delooping level and the derived delooping level of $B=\operatorname{End}_A(T)$. We show that the big finitistic dimension of $B^{\mathrm{op}}$ is bounded above by the derived delooping level of $\operatorname{Fac}T$ relative to $T$. Moreover, we prove that a self-orthogonal $τ$-tilting module $T$ is a $1$-tilting module whenever $2\text{-}\ddell_B(DT)$ is finite.

math.RT↗

arXiv2026

Derived Delooping Levels of One-Point Extensions and Finitistic Dimensions

Let $B$ be a finite-dimensional algebra, $M\in\modu B$, and $A=B[M]$ the one-point extension. We give a stable two-step extension criterion for the new simple $A$-module to have derived delooping level at most one. For a concrete algebra $B$ with a self-injective Nakayama quotient $R$, we determine the derived and classical delooping levels of $B[M]$ for every nonzero $M\in\modu R$. In particular, we obtain an explicit algebra $A$ satisfying \[ \Findim(A^{\mathrm{op}})=1<2=\ddell A=\dell A, \] which gives a negative answer to a question of Guo and Igusa.

math.RT↗

arXiv2026

Reduction techniques for the derived delooping levels

The derived delooping level is a recently introduced homological invariant that provides an upper bound for the finitistic dimension of the opposite algebra. In this paper, we employ two reduction techniques-cleft extensions and recollements-to study the finiteness of the derived delooping level of finite-dimensional algebras over a field. By applying the theory of cleft extensions to bound quiver algebras, we establish arrow-removal operations that preserve the finiteness of the derived delooping level. In parallel, using recollement techniques, we develop vertex-removal operations with the same finiteness-preserving property. We conclude with several examples illustrating the applicability and effectiveness of these reduction methods.

math.RA↗

arXiv2026

A characterization of IE-closed subcategories via $τ$-tilting theory

Enomoto and Sakai classified functorially finite IE-closed subcategories over hereditary algebras in terms of twin rigid modules. Their approach uses the hereditary assumption essentially and therefore does not extend directly to arbitrary finite-dimensional algebras. In this paper, we introduce canonical twin support $τ$-tilting modules and prove that, for an arbitrary finite-dimensional algebra, they are in bijection with left-and-right finite IE-closed subcategories, namely those whose generated torsion and torsion-free classes are both functorially finite. We further give a characterization of canonicality via the torsion-pair decompositions associated with $\operatorname{Fac} M$ and $\operatorname{Sub} N$, which yields a canonicalization procedure whenever the associated IE-closed subcategory is left-and-right finite. We also introduce canonical Ext-pairs. If the algebra is hereditary or $τ$-tilting finite, then functorially finite IE-closed subcategories are in bijection with isomorphism classes of canonical Ext-pairs, where the corresponding pair is given by the basic Ext-progenerator and the basic Ext-injective cogenerator. In the hereditary case, this recovers the twin rigid classification of Enomoto and Sakai.

math.RT↗

arXiv2026

Projectively Wakamatsu Tilting Modules over One-Point Extensions

Let $Γ= Λ[M]$ be the one-point extension of an algebra $Λ$ by a $Λ$-module $M$. We establish a method to lift projectively Wakamatsu tilting (PWT) modules from $\mathrm{mod}\,Λ$ to $\mathrm{mod}\,Γ$ by adding the new projective module, and prove that this lifting process perfectly preserves mutation relations under certain homological conditions. Furthermore, for source point extensions of representation-finite algebras, we obtain a complete classification of PWT $Γ$-modules in terms of those over $Λ$. In particular, we establish a bijection \[ \mathrm{PWT}(Γ) \longleftrightarrow \mathrm{PWT}(Λ) \coprod \mathrm{RPWT}(Λ, S_i). \] which yields the counting formula about $|\mathrm{PWT}(Γ)|$.

math.RT↗

arXiv2025

Some functors preserving exceptionality

We constructed some tensor functors that send each exceptional sequence in a module category to another exceptional sequence in another module category by using split extensions and recollements.

math.RT↗

arXiv2025

On the Gorensteiness of string algebras

In this paper, we give a description of the self-injective dimension of string algebras and obtain a necessary and sufficient condition for a string algebra to be Gorenstein.

math.RT↗

arXiv2024

Recollements and Gorenstein projective modules for gentle algebras

Let $A={\rm \mathbb{k}}Q/\mathcal{I}$ be a gentle algebra. We provide a bijection between non-projective indecomposable Gorenstein projective modules over $A$ and special recollements induced by an arrow $a$ on any full-relational oriented cycle $\mathscr{C}$, which satisfies some interesting properties, for example, the tensor functor $-\otimes_A A/A\varepsilon A$ sends Gorenstein projective module $aA$ to an indecomposable projective $A/A\varepsilon A$-module; and $-\otimes_A A/A\varepsilon A$ preserves Gorenstein projective objects if any two full-relational oriented cycles do not have common vertex.

math.RT↗

arXiv2024

Quotients of extriangulated categories induced by selforthogonal subcategories

Let C be an extriangulated category. We prove that two quotient categories of extriangu?lated categories induced by selforthogonal subcategories are equivalent to module categories by restriction of two functors E and Hom, respectively. Moreover, if the selforthogonal sub?category is contravariantly finite, then one of the two quotient categories is abelian. This result can be regarded as a generalization of Demonet-Liu and Zhou-Zhu.

math.RA↗

arXiv2024

Silting objects and torsion pairs in comma categories

In this paper, we first give a characterization of silting objects in the comma category Assume that C1 and C2 are two subcategories of left R-modules, D1 and D2 be two subcategories of left S-modules. We mainly prove that (C1, C2) and (D1, D2) are torsion pairs if and only if the induced two pairs in the special comma category are torsion pairs under certain conditions.

math.RA↗

arXiv2022

Gorenstein projective objects and recollements of Abelian categories

In this paper, we study the relationship of Gorenstein projective objects among three Abelian categories in a recollement. As an application, we introduce the relation of $n$-Gorenstein tilting modules (and Gorenstein syzygy modules) in three Abelian categories. For a recollement of Abelian categories, we show that a resolving subcategory induce two resolving subcategories. On the other hand, we also prove that two resolving subcategories can induce a resolving subcategory. Moreover, we give the size relationship between the relative global dimensions of three Abelian categories.

math.CT↗

arXiv2022

Balanced pairs and tilting modules in recollement

In this paper, firstly, we mainly study the relationship of balanced pairs among three Abelian categories in a recollement. As an application of admissible balanced pairs, we introduce the notion of the relative tilting modules, and give a characterization of relative tilting modules, which similar to Bazzoni characterization of n-tilting modules [4]. Finally, we mainly consider the relationship of relative tilting modules in a recollement.

math.CT↗

arXiv2022

Star modules with respect to balanced pair

n this article, firstly, we introduce the notion of star modules with respect to a balanced pair and obtain some properties. We mainly give the relationship between n-X star modules and n-X tilting modules [9], and a new characterization of n-X tilting modules.

math.CT↗

arXiv2022

Relative AR-correspondence, co-t-structure and silting pair

As a generalization of tilting pair, which was introduced by Miyashita in \cite{YM}, the notion of silting pair is introduced in this paper. The authors extends a characterization of tilting modules given by Bazzoni \cite[Theorem~3.11]{BS} to silting pairs, and proves that there is an one-to-one correspondence between equivalent classes of silting pairs and certain subcategories which satisfy some conditions. Furthermore, the authors also gives a bijection between equivalent class of silting pairs and bounded above co-t-structure.

math.CT↗

arXiv2021

Gorenstein syzygy objects in extriangulated categories

We give the definition of Gorenstein syzygy objects in extriangulated categories and obtain a characterization. For a recollement of extriangulated categories, we mainly show that Gorenstein syzygy objects induce a new Gorenstein syzygy objects under certain conditions, and prove that Gorenstein syzygy objects induce a new Gorenstein syzygy objects under certain conditions.

math.KT↗