arXiv subjects

Dajun Liu

Publications and source records attributed to Dajun Liu.

12 recordsLinked to original sources

arXiv2026

Projectively Wakamatsu Tilting Modules over One-Point Extensions

Let $\Gamma = \Lambda[M]$ be the one-point extension of an algebra $\Lambda$ by a $\Lambda$-module $M$. We establish a method to lift projectively Wakamatsu tilting (PWT) modules from $\mathrm{mod}\,\Lambda$ to $\mathrm{mod}\,\Gamma$ 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 $\Gamma$-modules in terms of those over $\Lambda$. In particular, we establish a bijection \[ \mathrm{PWT}(\Gamma) \longleftrightarrow \mathrm{PWT}(\Lambda) \coprod \mathrm{RPWT}(\Lambda, S_i). \] which yields the counting formula about $|\mathrm{PWT}(\Gamma)|$.

math.RT↗

arXiv2026

A characterization of IE-closed subcategories via $\tau$-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 $\tau$-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 $\tau$-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.

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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.

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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.

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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.

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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.

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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.

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